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q and if w > p, w > q
imply w > v (if v < p, v < q and if w < p, w < q imply w < v). In an analogous way we define the supremum (infimum) of any subset A of P, which
1.2 Notation and terminology
7
we denote by sup A (inf A). Most of the examples considered in this book are lattices, that is, posets P in which p v q and p A q exist for all p, q E P. Further, almost all posets that we will study are ranked posets, that is, posets together with a rank function. Here a rank function of a poset P is a function
r from P into the set N of all natural numbers such that r(p) = 0 for some minimal element p of P and p < q implies r(q) = r(p) + 1. Note that we do not suppose - as traditionally - that r (p) = 0 for all minimal elements p of P. If in a ranked poset every minimal element has rank 0 and every maximal element has the same rank, we speak of a graded poset (note that in any poset there is at most one rank function with this property). Given a ranked poset P, rp denotes throughout its rank function, but generally we omit the index P and merely write
r. The number r(P) := max{r(p) : p E P} is called the rank of P (note the difference from the weight w(P) of a weighted poset (P, w), which we defined by w(P) :_ EpEP w(p)). Very often we set for the sake of brevity n := r(P). A subset F of a graded lattice is called t -intersecting (t-cointersecting) if r (p A q) > t (r(p V q) < r(P) - t) for all p, q E F. Intersecting (cointersecting) is an abbreviation for 1-intersecting (1-cointersecting). The dual of a ranked poset P is the dual P* of P together with the rank function r p* := rp(P) - rp(p) for all p E P. Moreover, the product of two ranked posets P, Q is defined to be the poset P x Q together with the rank function rp,< Q given by rpxQ(p, q) := rp(p) + rQ(q). For a ranked poset P, we define the ith
level by N,(P) := {p E P : r(p) = i}; its size W1(P) := 1N;(P)I is called the ith Whitney number, i = 0, ... , r(P) (when there is no danger of ambiguity,
we write briefly Ni and W,). It is useful to define Ni := 0 and W; := 0 if i V {0, ... , r(P)}. Obviously, each level of a ranked poset is an antichain, and the union of k levels is a k-family. The rank-generating function F(P; x) of a ranked poset is defined by F(P; x) JpE p xr(P)(_ _i o) W,x`). It is easy to see that
F(P x Q; x) = F(P; x)F(Q; x) if P and Q are ranked. For S c {0, ... , r(P)}, we define the S-rank-selected subposet (PS, ws) as the subposet induced by PS JP E P : r (p) E S} together with the induced weights ws. For ranked posets P, Q of the same rank, we define the rankwise (direct) product P Xr Q to be the set Ul ( Ni (P) x Ni (Q) together with the relation (p, q) < p x,. Q
(p', q') if p
p} (resp. /(p) := {q E P : q < p}). More generally, if P is ranked, let the upper (resp. lower) k-shadow of p be defined and
denoted by V.k(p) :_ {q E Nk : q > p} (resp. A- k(p) :_ {q E Nk : q < p}).
Introduction
8
The (k-) shadows of a subset of P are the unions of the (k-) shadows of its elements. More generally, given a weighted and ranked poset (P, w), the weight w(N,) of the ith level Ni of P is called the weighted ith Whitney number. We mostly use the following definitions in the w - 1 case (where W1 = w(N1)). The weighted and ranked poset (P, w) is said to have the k-Sperner property if the maximum weight of a k-family in (P, w) equals the largest sum of k weighted Whitney numbers in
(P, w), that is, if
dk(P,w)=max{w(N;,)+...+w(N,,):0 3, only asymptotic bounds and estimates are known (see Poljak and Tuza [383], and Gargano, Ko"rner, and Vaccaro [216, 217]). As an example we study the following condition: Two ordered 3-partitions (XI, X2, X3), (YI, Y2, Y3) of [n] are called KS-independent if XI n Y3, X2 n Y3, YI n X3, Y2 n X3 # 0. Let KS(n) be the maximum size of a family of pairwise KS-independent ordered 3-partitions of [n]. The following result is due to Ko""rner and Simonyi [314]:
Theorem 2.1.5. We have
KS(n) =
Proof. First we show the inequality ">n
(k)
otherwis e.
Both bounds are the best possible. Proof. First observe that (a) and (b) are equivalent since .T is It -wise intersecting
if T (the complementary family) is µ-wise cointersecting. Thus it remains to prove (b). The bound cannot be improved since {X E (m)
:
1
X}
ifk > µ,
.T*
D']) k
ifk
then d
_ (12 ... n) and X* = {1, ... , k}. Let
Proof of Claim. Suppose again that this time S:=
E [ n ]: { i , i + l ,
. . .
,i+k-1}E
(2.21)
ISI.
Note that 1 E S. We partition [n] into k sets S1, ... , Sk, namely, the residue classes modulo k. We show that
ISj n Sl < ISjI - 1 for all j E [k].
(2.22)
Assume the contrary, that is, Sj = S n Sj for some j E [k]. Then (with addition
modulo n) the sets XI := { j, j + 1, ..., j + k - 1}, X2 := (j + k, ..., j + 2k - 1}, ..., Xa := t j + (a - 1)k, ..., j + ak - 1} would belong to .T', where j + (a - 1)k < n, j +ak - 1 > n. From the first inequality we derive a - 1 < k that is, a < µ. If ak > n, then Ua1=1X1 = [n] in contradiction to the fact that F is µ-wise cointersecting. But if ak < n then a < µ, hence a + 1 < t and we have Ua,=I XI U X* = [n], which is again a contradiction. From (2.21) and (2.22) we obtain k
k
nSjI 4. Then
ifk n + 1. Case 1. The three members Xa_l, Xa, Xa+1 belong to the chain C' of type (2.23). If n + 1 V Xa+1, then the three members belong to the chain C of ("(n), and we can find by the induction hypothesis a corresponding chain D E (E(n) with the corresponding member Xa which also belongs to the chain D' E (E(n + 1).
Since IDI = ICJ - 2, also ID'J = IC'I - 2. If n + 1 E Xa+l, then Xa = Xh, and we can take la := Xh_1 U In + 1}. The corresponding chain is C". Case 2. The three members Xa_1, Xa, Xa+t belong to a chain C" of type (2.24). Then Yj := Xj - in + 1}, j = a - 1, a, a + 1, belong to C E (t (n), and we find, by induction, a corresponding chain D E t(n) with the corresponding member Ya. Because Ya+1 cannot be the maximal element of C, the member Ya is not the maximal element of D (otherwise we would have I D I < ICI - 2).
Thus Xa := Ya U in + 11 belongs to D", and Xa is the desired element since
ID"I=IDI-I=ICI-3=IC"I -2. We order the members of Q(n) by increasing size; that is, (E(n) = {Cj :
1<j
U(n) := max{i(.F) : F is a filter in 2Eu3}.
Extremal problems for finite sets
32
The number tlr (n) is sometimes called the Shannon complexity of the problem of the recognition of monotone Boolean functions. It was studied the first time by Korobkov and Reznik [316]. The following result is again due to Hansel [252], who improved an earlier estimation of Korobkov [315]. Theorem 2.2.3.
We have *(n) = (l"_Ti_) + (l'i).
Let J :_ {X c [n] I X I > ["21 J I. Obviously, for all X c [n] with M = L"21t J, also F* - [X] is a filter, and, for all X C_ [n] with Proof.
:
IXI = L"21 J, .2 U {X} is a filter. Thus, knowing the Heidrun answers for all Sebastian cannot determine the Y E 21"1 - {X), where X E (L [n; j) U filter uniquely. Consequently, he must ask eidrun about all these X. " k, since one member of F1 already produces a k-element shadow. Thus the claim yields the contradiction I.F2'1 > k.
Casel.2.k+1 <x.WehaveFil = IFI-IF21 > I < IF, I < IFI we may apply the induction hypothesis to obtain I0 (F1) I and again the claim yields the contradiction I.F2'1 > (k-i). Case 2. I.F2'1 (x-11). By the induction hypothesis, Io(F2)I > (x-2'). Hence
IL(T)1=1O(.Fz)I+IJ721-(k-2)+(k-1)=(kz
1).
The preceding proof is due to Frankl [191]. Let us look at some consequences. Repeated application of Theorem 2.3.1 gives the following estimation:
2.3 Exchange operations and compression
Corollary 2.3.1.
37
Let F be a k-uniform family in 2In] With I FI = (k), where
k<x ." Let T* := {X c [n] : 1XI < m}. Obviously, for all X E 1* U X is an ideal; moreover, I* - (m'I,) is an ideal. Thus Jana must ask Heidrun about all elements of ([Mn) and about at least one element of ([nil) in order to be sure. " I B(9) I Case 2. 91 (i) U B'(91 (i)) D 9'2(i). Then
IB(9)I = IB'(91(i))I + IB'(92(i))I. Clearly, however,
IB(F)I > IB'(.Fl(i))I + IB'(.EZ(i))I, and again the induction hypothesis yields I B(F) I > I B(9) I From this claim we derive that
XEF,Y<X,Y#XimplyYEF
(2.33)
since, if Y X, there is an i X U Y or a j E X n Y (.171(i) (resp..F2(j)) are compressed). Now we assume that .F ¢ C(m, 2N). Then there must be a set A C [n] with A 0 F for which the next set E (with respect to IEin(F)I, IB(Q)I = IB(.F)I - Ln
< IB(F)I
2
This is a contradiction to the choice of F. As we have seen, the sets Sm := C(m, 2N)
are solutions of the EIP and MEP (resp. VIP). Obviously (we used this fact already), SO
We say that there is a nested structure of solutions (NSS) (which need not exist for other graphs). The existence of an NSS of the EIP for a graph G = (V, E) enables the solution of an optimal numbering problem: Let I V I = n. We wish to minimize
aG(l) := > I1(w) - l(v)I vwEE
over all bijective mappings 1 : V -- [n].
Theorem 2.3.4. Let So c . . . c Sn be an NSS for the EIP of G = (V, E), and let v, be the (unique) element o f Si - S i _ 1 , i = 1, ... , n. Define the bijective mapping
1*:V
[n]byl*(vi):=i,i = I__ n. Then aG (1 *) < aG (1) for all bijective mappings 1 : V -* [n].
Proof. Let 1 : V -+ [n] be bijective. We orient the edges of G such that 1(e+) > 1(e-). This leads to a directed graph G = (V, E). Let T, := {v E V : 1(v) < i}. We have
aG(1) = 1: 1(e+)-1(e ) eeE
n-I
_
1= eEE i:1(e-)ak> >ai+i >x>1.Then, for0
(i)++ (l+l+i-k ak
+
k,
x
a[+i
+
l+ i-k
We encourage the reader to prove this corollary (for i = k - 1) directly by the proof method for Theorem 2.3.1. Katona [292] found the Kruskal-Katona Theorem when solving a problem of Erdo"s:
Theorem 2.3.7. Let F = {Xi , ... , X,,, } be a k-uniform family in 21"I , 0 < i < k, and let
m*:-(k k')+(k k' 12)+.(kkZ 24) +... I f 0 < m < m * then there exist distinct sets Y 1 ,
Y,,, in
{
(k-i).
(M) such that Yj c Xj
for all j. The bound is the best possible. Proof. Because of Hall's theorem (Theorem 5.1.2) we only must show that
IA-,i(T) I ? IF'I for each subfamily F' of Y. Since I.T" I < I FI < m *, there exist an integer 1 E {k - i, ... , k} and a real x with
1 <x (
k+i i
)+
k+i -2 +...+ 21+2+i -k (k+
i-1
x +l+i-k
1+1+i-k
Note that ( ') > (U) for 0 < u < v < x < u + v, which can be checked in a straightforward manner. Hence
Io
? (i+_) - ()
'-0.
The bound is the best possible since the i-shadow of C(m* + 1, (kl)) is obviously smaller than m * + 1.
Leck [333] studied the problem of minimizing the size of the i-shadow of F = {XI, ... , Xm } c_ ("') for m > m* under the supposition that there are sets Yl, ... , Ym E (nl) such that Yj C Xj for all j.
2.4. Generating families In the last decades several methods for proving theorems on families satisfying a certain intersection condition have been worked out. We have seen some of them in the previous sections. We will not discuss those methods that yield the maximum size of the families under consideration if n is sufficiently large. We refer here only to Frankl [189] and Schmerl [416] who built up a theory that is based on a theorem of Erd6s-Rado [171]. The powerful method that is described in this section is very new. It was developed by Ahlswede and Khachatrian [15]. Though it is much more far-reaching, we will restrict ourselves to the highlight in extremal set theory: the complete determination of the maximum size of k-uniform t-intersecting families by Ahlswede and Khachatrian [15].
Let F C
(lkl)
A family 9 C 21n1 is called a generating family for F if . V +k (G) = F. Let O (JI) be the class of all generating families for F. Note that F E 15 (F ). We will study generating families for special families. A family .F is called left (resp. right) shifted if
sij (F) = F for all 1 < i < j < n (resp. for all 1 < j < i < n). We may also define these families in an equivalent way as follows: On 21n] we consider the relation -s which is the reflexive and transitive closure of the relation Ul< j_X,.. addition, we introduce for X E 2h1, F c 2[n] the number
51
i. In
max(F) := max max(X) = max maxi. XE.F
XE.F icX
Moreover, for F c (k), let
ak(F) := min max(Q). BEe5(.F)
Thus, ak(.F) denotes the smallest number i such that there is a family Q c 21il which is generating for .T'.
Proposition 2.4.2. Let .F c (kl) be left shifted. Then there is a left-shifted family Q that is generating for F such that max(Q) = ak(F ).
Proof. Under all families Q E Q5(.17) with max(g) = ak(.T ), take one of maximum size. This family Q is left shifted. Assume the contrary. Then there are X Q, Y E Q such that for some 1 < i < j < n, X = di j (Y) where i Y, j E Y. If we can show that V +k(X) c F, we obtain the desired contradiction, since then Q U {X} is also generating for F (recall that Q is of maximum size). So let X CZ E (kl). If j E Z then Y C Z, which implies Z E F since Y E Q; that
is, V k(Y) c Y. If j V Z, let W := 1 (Z). It is easy to see that Y C_ W E F. Because F is left shifted, it follows that Z = aid (W) E F. Since with each family Q, also the (Sperner) family Q* of minimal elements of is generating for . F, we derive from Proposition 2.4.2:
Proposition 2.4.3. Let F c (lkl) be left shifted. Then there exists a Spernerfamily that has the following properties:
(a) Q* is generating for F,
(b) max(g*) = ak(-F), (c) if Y <S X and X E Q* then there exists some Z E Q* such that Z C Y.
We call a family Q* that has the properties described in Proposition 2.4.3 an Ahlswede-Khachatrian family (briefly AK family) for .T'. For X E g*, let
V'' k(X):={YE I [k}I:XcY and Y-Xc{max(X)-I-1,...,n}}, and, as usual, we define
UXEG o',k(X)
Proposition 2.4.4. Let T- c (m) be left shifted and Q* an AK-family for Y. Then the sets V.' k (X), where X E Q*, partition the family Y.
Extremal problems for finite sets
52
Proof. Since Q* is generating, 0' k (X) C_ F for all X E G*. First we show that all members of F are covered. Let Z E Y. Since Q* is generating, there is some X E C* with Z E V_,k(X). Under all such sets X we take one for which I XI is minimal and, in second instance, J (Z - X) fl { 1, ... , max(X)JI is minimal. It is enough to show that this intersection is empty. Assume there
is some i E (Z - X) n (1..... max(X)}. Note that i < max(X). Let Y :_ 2k - t since for n < 2k - t every k-uniform family is automatically t-intersecting. As a candidate for a maximum family we have
So:={XE
(1)
[t]
X}.
Erd3s, Ko, and Rado [170] have already shown that So is indeed a solution if n is
2.4 Generating families
53
sufficiently large. But there are other candidates, namely the families
Sr:={XE (In]):IXn[t+2r]I >t+r},
r=0,...,k-t
which are easily seen to be t-intersecting. Frankl [188] conjectured that for any n the "best" of these families Sr is the solution. So let us first determine the best family Sr. Lemma 2.4.1.
We have
t+2r
r
(a) ISr)
t+r+i)(k
(b) ISri < (resp. =)ISr+l l
n-t-2r t
r
iff n < (resp. _) (k - t + 1) 2+ r + 1).
Proof. (a) Sr is the disjoint union of the sets {X E (n])
:
IX n It + 2r]I
t +r + i),i = 0,...,r. (b) It is easy to see that
Sr+l - Sr = {X E ([k]) : ix n [t + 2r]I = t + r -
{t+2r+1,t+2r+2} C X}, Sr - Sr+1 = {X E ([k]) : I X n It + 2r]I = t + r,
{t+2r+ 1,t+2r+2)nX=0}. Thus ISr I < (resp. _) ISr+l I is equivalent to the following relations:
Sr I- Sr+11
< (resp.
ISr+1 - SrI
n-t-2r-2 k-t-r t+2r(t+r-,k-t-r-1) /n-t-2r-2 t+2r
t+r
)
< (resp.
n < (resp.=)
(k-t+1)2+r+1)
In particular it follows for n > (k - t + 1) (t + 1) that ISO I ? IS, I > IS21 > Frankl [188] (for t > 15) and Wilson [470] (for any t) proved that So is indeed a maximum k-uniform t-intersecting family if n > (k - t + 1)(t + 1). We will present Wilson's algebraic approach in Section 6.4. Some further special cases of Frankl's conjecture were settled by Frankl [188] and Frankl and FUredi [196]. But the complete solution is due to Ahlswede and Khachatrian [15]:
Extremal problems for finite sets
54
Theorem 2.4.1 (Complete Intersection Theorem). Let 2k - t. Let r E {0, ... , k - t } be that number for which
1 < t < k < n, n >
(k-t+1)(2+r+1) > t,
(b) IXInX21 > t+1 ifthereare i, j E [n]withi < j, i
XIUX2, j E XInX2.
Proof. (a) Assume that there are X1, X2 E G* with i XI n X2I < t - 1. For the set A :_ [n] - (XI U X2), we have
JAI =n-IXIUXi1 >2k-(t-1)-IXIUX21
= k-IXII+k-IX2I+IXInX2l-(t-1). Thus we find in A two subsets Y1, Y2 with I Yi l = k - I Xt 1, l = 1, 2, and I YI n Y21
k-(n-t-2r-S),1=1,2. Proof of Claim 1. By Lemma 2.4.3(a), I Xl n X21 < t can hold only if X1, X2 E
Q'2and(XIU{t+2r+8})n(X2U{t+2r+3})=t,thatis,IX1nX2I=t-1. But Lemma 2.4.3(b) implies that there is no i < t + 2r + 8 with i 0 X1 U X2. Thus
IXll+1X21 = IX1nx21+IXiuX21
= t- 1+t+2r+S- 1 =2t+2r+S-2. If, for example, 1X1 I < k - (n - t - 2r - S) - 1, then IX2I > n - k + t - 1 > k (recall n > 2k - t). This is a contradiction since obviously IX2I < k - 1. Now we classify 92 and g' with respect to the size of the elements (and use a new letter):
7-t, :={XE92: IXI =i},
f; :={XE9':IXI =i-1}
(thus we make an exception from our general rules: The members of 7-l' have size i - 1, but note that they can be obtained from the members of 7.1, by deleting the
element t + 2r + 8). Clearly we may restrict i to t < i < t + 2r + S. If Ht 0, then 191 = 1 since g is a Sperner family that is by Lemma 2.4.3(a) t-intersecting. From Proposition 2.4.3(c) it follows that 9 = {[t]}, in contradiction to (2.43).
Extremal problems for finite sets
56
0, then 92 = {[t + 2r + S]} = 9 because g is a Sperner family and max(g) = t + 2r + S. Since F has maximum size, necessarily 8 = 0, in If 7-lt+2r+a
contradiction to (2.43). Thus we may suppose that
U
92 =
7-1,
and
92 =
t oo.
69
2.5 Linear independence
Proof. Choose any c > 0. We have to show that there exists some no such that p > (1 - e)n for all n > no. By Theorem 2.5.8, 7r(n)
n((1 - E)n)
ti
n
logn + log(l - E)
(1 - E)n
logn
1
if n is sufficiently large. Thus there exists a prime in the interval ((1 - E)n, n].
The diameter of a finite set of points in the n-dimensional Euclidean space Rn is the maximum distance between any two points of the given set. Let bd (n) be the smallest number such that every finite set in 1[l;" of diameter d can be partitioned into bd(n) sets of diameter smaller than d. Using contractions one can easily see
that bd(n) does not depend on d, so we write briefly b(n). Moreover, it is left to the reader to show that b(n) is finite. Taking the vertices of a regular simplex in Rn one immediately obtains b(n) > n + 1. In 1933 Borsuk [79] conjectured that equality holds (also for infinite, closed sets). Not until sixty years later did Kahn and Kalai [281] kill this conjecture and show that the lower bound can be significantly improved:
Theorem 2.5.9.
We have b(n) ti
as n - oo.
Proof. First we prove the relation for a subsequence of the natural numbers. Note that b(n) is clearly increasing. Let p be a prime (tending to infinity) and let m := 4p. We will construct a set T of points in the m2-dimensional Euclidean space of diameter d := 12-m. Let S be the set of points x = (xi, ..., xm)T for which xi E {-1, 1} for all i, xl = 1, and the number of negative components
equals z . Obviously, ISI = 2 m. With each x E S we associate the set X {i E [m] : x; = 1) E (ll). For any x, we put
x *x :_ (XIXI,XIx2,...,xlxm,x2xl....,x2xm....,xmxm)T
Now let T{x*x:xES}.Since x*x/ y*yifx:y, ITI = 2I
m)
(2.77)
2
In order to compute distances, we must be able to calculate the standard scalar product (, ). In a straightforward way one may verify the identity
(x * x, y * y) = (x, y)2.
It is easy to see that for x, y E S and the associated sets X, Y (with
[m]-X,Y:=[m]-Y)
(x, y) = IXnYI+IxnYI-(IX-YI+IY-XI) = m - 2IXI - 2IYI + 4IXfl YI;
(2.78)
Extremal problems for finite sets
70
that is,
(x,y)=41XnYI-m.
(2.79)
Thus we have for the distance p between two points x * x and y * y of T :
p2(x *x,y *.Y) = (x,x)2+(y,y)2 -2(x,y)2
=
m2 +M 2
- 2(41x n Yj
- m)2 < 2m2,
and equality; that is, the diameter d = 12-m is attained iff I X fl Y I = a = p. Let U be a subset of T of diameter smaller than d and let r be the family of those sets X that are associated with the vectors x for which x * x E U. From the preceding discussion we know that .T' is a 2p-uniform family in 21' with the p - 11. Consequently, property given in Theorem 2.5.7, where L
ICI
;=p
p
p
p
+ im-p+m-p+lm-p+2 (mp/ l
(M)
b(n) N (1.203) ' ti (1.2)'
as n -+ oo.
The construction is a reformulation of that given by Kahn and Kalai [281 ]. For the computations, we used ideas of Nilli and his best friend Alon [378], who also
2.6 Probabilistic methods
71
noticed that one can enlarge the set S by taking those x for which the number of negative components is even (instead of ). 2 with which the points in I[8" can be Let cd (n) be the smallest number of colors colored such that no two points of the same color have distance d. Again, cd(n) does not depend on d, so we write c(n). It is an easy exercise to show that c(n) is finite.
Corollary 2.5.2.
We have c(n) ti (1.2) " as n
oo.
Proof. Take a set of points of diameter d that cannot be partitioned into less than b(n) sets of diameter smaller than d. Every "good" coloring (with respect to the distance d) of this set needs at least b(n) colors since the color classes are sets of diameter smaller than d. Thus (by Theorem 2.5.9)
c(n) > b(n) Z (1.2)'In- as n --> oo.
This result is due to Frank! and Wilson [202], who proved a conjecture of Larman and Rogers [332] more than ten years before Kahn and Kalai [281] found Theorem 2.5.9. The original proof of Frank! and Wilson is similar to the proof of Theorem 2.5.9, only the construction is easier.
2.6. Probabilistic methods Our general aim is the maximization of the size of families satisfying certain conditions. For example, for Sperner families and k-uniform intersecting families we could first "guess" a maximum family of the kind in question, and then prove
that these families are optimal. But sometimes we do not have an idea how to produce such a large family. Here it is useful to apply the probabilistic method, sometimes also called the Erdds method in honor of P. Erdos, who not only is the founder of this area but also developed it with numerous deep results and still more intriguing problems. This method can be described as follows: In order to prove
the existence of a family of "large" size satisfying the given condition, one has to construct an appropriate probability space and to show that a randomly chosen element (corresponding to a family) in this space has the desired properties with positive probability. Only a few instructive examples are presented here together with modifications of the method. The interested reader may learn much more in the monographs of Erd6s and Spencer [172] and Alon and Spencer [29]. Certainly, also other facts from probability theory are useful in combinatorics. One example
is the entropy inequality (Theorem 2.6.5). The application of limit theorems is studied in Chapter 7, where some standard definitions and results can also be
Extremal problems for finite sets
72
found. Working in this area, one must be clever in calculus; in particular, it is useful to have many inequalities at hand. Here is an important one: Proposition 2.6.1.
(a) For all x E IR, 1 + x < ex, and equality holds only for x = 0.
(b) For ally > 0, logy < y - 1, and equality holds only for y = 1. Proof. (a) The function f (x) := ex - x - 1 has a unique minimum at x = 0. (b) By (a), y < ey-1; that is, for y > 0, logy < y - 1. Our first example is due to Kleitman and Spencer [311] and also presented in the survey of Alon [27]. A family F in 21"] is called k-independent if, for every k distinct members X1, ... , Xk of F, all the 2k intersections nk_1 Y; are nonempty where each Y; is either Xi or Ti. Theorem 2.6.1. The maximum size of a k-independent family in 21"3 is at least [Zkll/ken1(k2k)J for all k > 2.
Proof. Let, for X c [n], X1 : = X and X0 := X. Consider the random family F = {X1, ... , Xm}, where (X1, ... , Xm) is a sequence of m randomly, independently chosen subsets of [n] and equidistribution is assumed (i.e., the probability of the choice of any particular subset equals 2-" ). More precisely, the elements of our probability space are the n x m-arrays whose entries take on only the values 0,1, and the (classic) probability of such an array equals 2-"m . The columns of the array are the characteristic vectors of the members of Y. Note that up to now the members are not necessarily different. We have P(F is not k-independent) < P({Xi...... Xjk} is not k-independent) 1
+2
(_/))
0.5204 * 2n.
n
(2.80)
We will see that for sufficiently large n there exists a Sperner family F in 21n1 such that
I0(Jc)I>0.1146*IQI.
(2.81)
Together with (2.80), this yields for sufficiently large n,
IA(F)I > 0.059*2n.
Let t := u + 1 -1, c := e(log(e + 1) - 1) = 0.8515... , and p be the solution of
tp(k) = c. Clearly, limn +
(k) = oo; thus,
lim tp = 0.
(2.82)
n-*oo
Let l; be a random variable taking on values from S
p
{1,
.
. .
, t, t + 1} such that
if i = 1,...,t,
1-tp ifi=t+1. Let {I;K : K E (1k)} be a set of (completely) independent copies of 4 (in the
following K always belongs to (1k)). For j = 1 + 1, ..., u + 1, we define the
2.6 Probabilistic methods
75
random families
,Fj := {X E ([nI)
K c X) = j
:
'17:= U F. j=1+1
Note that then the random family A (.F) is contained in Q and that I A (F) J is a random variable which is a function of the i;K, K E (kl). Claim.F is a Sperner family in 2(").
Proof of Claim. The existence of some j, j' with l + 1 < j < j' _< u + 1 and of some X E Fj, X' E .Pj, such that X C_ X' could be the only obstacle. Since X E F j, there is some K C_ X such that K = j - 1. But then also K C X' and min 14K : K C X'} < j -1 < j' -1 in contradiction to X' E Jj'. For each A E Q, we define the random variable 1)A by 1
71A =
10
ifAEA(,r), if A 0(F).
Then
E(Io(F)U = E
E
\AEQ
71A)
_
P(A E 0(F)).
E(11A) = AEQ
AEQ
If we can show that P(A E .(F)) > 0.1146 for all A E Q, then there must be a realization of the K, K E ((k)), that is, also a realization of F, such that (2.81) holds.
Claim. P(A E A(F)) > 0.1146 if A E Q. Proof of Claim. For sake of abbreviation, always let i A C [n], JAI = j, I < j < u. Then
j + 1 - 1. Let
P(A E 0(F)) = P(3b E [n] - A : min{l;K : K C A U {b}} = i)
> P(min{K:KCA}>iand 3bE[n]-A:min{I;K:KCAU{b}}=i) = P(min{'K:KCA}>iand 3bE[n]-A:minRK'u(b):K'C A, K'E(k")l)) =i). In the following, K' always denotes an element of ( kn)1). The events in the last conjunction are independent since the index sets of the corresponding random variables, namely {K : K C A} and (K' U {b} : K' C A, b E [n] - A} are disjoint
Extremal problems for finite sets
76
and the K, K E ([k1) are (completely) independent. Thus we may continue:
K C A) > i)P(2b E [n] - A :
P(A E 0(.F)) =
: K' C A} = i).
u {b)
(2.83)
Now we compute and estimate both factors. We have (note (2.82))
i) = (1 - ip)(k) > (1 - tp)(j)
P(min{K : K C_ A} > i) = fl KcA
> (1 - tp)(k) = (1 - tp)`It' = e-`(1 + o(1)). For the second factor, we need a little bit more effort:
P(2b E [n] - A : min{4K'u{b} : K' C A) = i)
= 1 - P(Vb E [n] - A :min{4K'u(b) :K'CA}¢i)
= 1 - fl P(min{K'u{b}:K'CA)#i) bE[n]-A
=1- fl (1-P(min{i;K'u{b}:K'cA}=i).
(2.84)
bE[n]-A
Next we estimate for all b E [n] - A the probability in the RHS of formula (2.84). We have u (b) : K' C- A) = i if for all K' C_ A, K' u {b} > i and not for all K' C A, 4K' U {b} > i. Thus
P(min{ K'u(b) : K' C A) = i) _ (1 - (i -
(1 - ip)(k1)
> (1 - ip)(k'i)
>
(k-
1
P
(for the last estimation we used the Binomial Theorem and the relation 0 < 1-ip < 1 for sufficiently large n in view of (2.82)). For the first factor of the RHS, we have by Bernoulli's formula,
1 > (1-ip)(k''1)>
1-(k, 1)ip>
1-(k)tpu-k+l
(k u
l)tp
+i=1+0(1).
To estimate the second factor, we first observe
(k)
-
(U)] u
(u)('
(u-k+1)k
u-k+1 -1 u - k + 1)
(k) u-k+1
(k)
a-!
_
(uk)e-1 ( 1 + 0(1)).
(u-k+1)k
77
2.6 Probabilistic methods Thus
> J) p
(k
k
- (k) j - k + 1 p tp(k)t(j k+ 1)
1e-1(1 +o(1)) = 1(1 +o(1)),
and we derive
P(min {'K'U(b} : K' C A} = i) > el (1 +o(1)). Insertion into (2.84) yields
K' C A} = i)
P(3b E [n] - A :
> 1 - (I -
-
C
el
(1
+o(l))n-j
>1-e
ei(1+o(1))("-j)
C
e-e)(,+a(,))
(here we used Proposition 2.6.1(a)). Thus the estimation of both factors in (2.83) yields
P(A E A(.F)) > (1 +o(1))e-`(1 - e-e). For c = e(log(e+ 1) -1), the function on the RHS attains its maximum 0.1147... . Thus, for sufficiently large n,
P(AEA(.T'))>0.1146.
Without proof we mention the following upper bounds of Kostochka [322]:
Theorem 2.6.3. If n is sufficiently large then for any Sperner family Fin 2(nl (a) I A (F) I < 0.725 * 2",
(b) IA (F) U V(.P)I < 0.9994 * 2". A related result for the boundary of a filter was previously obtained by Kostochka [321]. The next modification of the probabilistic method for the construction of large families without forbidden configurations has the following basic idea (see, e.g., Kleitman [305]): Construct a collection (i.e., repetitions are allowed) of m members entirely at random. Then compute the expected number of forbidden configurations in it (e.g., comparable elements, nonintersecting members, etc.). If this number is at most km, then there must be some collection having no more than Am forbidden configurations, so that, by omitting at most elm members, we can obtain a collection (often automatically without repetitions, i.e., a family) of
Extremal problems for finite sets
78
the desired form of size (1 - A)m. Now let us illustrate this idea (with a slight modification). We are looking for large (p, q)-Sperner families in Bn (see [157]). Here a family .F of subsets of [n] is called a (p, q )-Sperner family if the intersection
of any p members of F is not contained in the union of any other q members of F. Note that a (1, 1)-Sperner family is a usual Sperner family. Let d"; p,q be the maximum size of a (p, q)-Sperner family in B". Theorem 2.6.4.
We have n
2p+q dn;p,q
C (2p+q
p+q-1
- 1)
P+9-1
where c:=(p+q-1) app+-+q Proof. Let (X1, ... , Xp,
... , Xp+q) be a sequence of p + q randomly, 1 ,--independently chosen subsets of [n], where equidistribution is assumed (analogous 2p+q to the proof of Theorem 2.6.1). Let a := 2p+q -1 Claim. The probability of the event X1 fl . . fl Xp c Xp+l U . . U Xp+q equals (Q
)n.
Proof of Claim. As in the proof of Theorem 2.6.1, we consider the random sequence as a random 0, 1 -array with n rows and p + q columns where the jth fl XP c Xp+1 U . . U column is the characteristic vector of X. We have X1 fl Xp+q if in the corresponding array there is no row of the form (1, ... , 1, 0, ... , 0) (2p+g-1 )" (p ones and q zeros). This gives the probability Zp+q Now we construct at random a sequence (X1, ... , Xm) of m subsets of [n] and after that we pick, step by step, p + q of them in all possible ways, which gives
m (m - 1) ... (m - p - q + 1) sequences of the form (Xi1..... Xip, Xip+l , ... , Xi p+9 ). By Claim 1 we obtain for the expected number µ of events Xi 1 fl ... fl X; p c Xip+1 U . . . U Xip+q the value
For.k > 0, we choose the largest m such that µ < Am. We then have 1
m > (.ka") p+g-1
since form < r(Aa") p+q-1 1
/ a)n
µ < m(m - 1)p+q-1 I
< m(,la")
(!)n
= ,lm.
Clearly there must be a fixed sequence (X1, ... , Xm) with at most Am "bad" events described above. We may destroy such an event by taking away, for example, X,1.
79
2.6 Probabilistic methods
Then simultaneously p!q! events are destroyed, namely those that are obtained by permuting it, ... , i p and 1p+1 , ... , i p+q, respectively. After all these events are destroyed, we retain a sequence whose members form a (p, q)-Sperner fam-
ily of size at least m - ``(note that repetitions always yield a "bad" event). Consequently, dn; p,q
where fo,) = (1 - ;' , )A v+a
n
f(A)a P+9-T {
The function f (,l) attains its maximum in R+ at
A = p+9 . The corresponding value of f equals the constant c from the assertion.
The concluding example of this section shows that other parts of probability theory also can be applied successfully. First we need some preparations. Let S again be an at most countable subset of R. Let = (v'1, .. . , ln) be a discrete random vector that takes on only values from S. Further let (i i, ... , 1Jr) be that random vector which takes on values from Sr with probabilities
P(771=Sl,...,rlr=sr)=
E
P(1=s1,...,4.=sn)
(Sr+l,...,S,,)ESf-r
Here we call this vector marginal random vector. Clearly, it can be considered as the restriction of the old vector to the first r components, so we denote also the new
vector by(1;1,...,lr)instead of(rll,...,>)r).For1 L
Proof. (a) Suppose, w.l.o.g., that [1, i] E.F. By the suppositions of the lemma every member of Jr must have a starting point or an endpoint in [ 1, i ]. Let S (resp.
E) be the set of starting points (resp. endpoints) of elements of Jr lying in [1, i]. In particular, 1 E S, i E E. Clearly, [1, i] is the only element of Jr having both
a starting point and an endpoint in [1, i]. Consequently, IJI =ISI +IEI - 1. The elements of E + 1 :_ {k + l(modn) : k E E} cannot be starting points i - (IEI - 1) (we cannot exclude i + 1) and by the suppositions. Thus ISI
IJI IJ7I + 1, there holds IVC (.F)I
>1+
IFI
> i+1
1
i
I.FI
Cases B=I and B=IvC. We treat these cases together. The proof is outlined for B = I ; the particularities
for B = I v C are attached in boxes. Step 2. The inequalities < 1 are true for all i < 2 Lemma 3.3.2(a) note that Fi is intersecting for i
0. Claim 1. If [1, m] E .T', then either [2, m + 1] or [m + 2, 1] is in T. Proof of Claim 1. Suppose that [m + 2, 1 ] F. We try to add [2, m + 1 ] to .T' (if it is not already there). The only obstacle could be a member X of F that is either
contained in [2, m + 1] or contains [2, m + 1]. In the first case, X has to contain m + 1, because otherwise it would be a subset of [1, m]. Thus X = [j, m + 1] for some 2 < j < m + 1. Similarly, if X contains [2, m + 1 ] then X = [2,1] holds for some m + 1 < 1 < n. Since F is an S-family, at most one of these possible sets X can be in F. Delete this X and add [2, m + 1] to F. This change increases the LHS of (3.4), which is a contradiction to our choice of F, and the claim is proved.
A pair of complementing, in C consecutive, m-element subsets is called an equipartition. We say that an equipartition is represented in F iff one of the parts is a member of F. Claim 1 states that if an equipartition is represented in F then the neighboring equipartition is also represented. By induction, this results in
3.3 Boolean expressions
103
Claim 2. All equipartitions are represented in .F; that is, fm = m.
Claim 3. Let X and Y be members of F with sizes different from m. Then X n Y 0. Proof of Claim 3. Let J Xi < J Y J and, w.l.o.g., Y = [1, j]. First suppose j < m. By Claim 2 and since F is an S-family we have [j + 1, j + m], [m + 1, n] E .T'. If X is disjoint to Y, then X must be contained in the union of [j + 1, j + m] and [m + 1, n ], but neither of these sets can contain it alone. Hence [m, j +m + 1 ] c X
follows. But then JXi > j, a contradiction. The case j > m is easier. The set [m + 1, n] (belonging to F, as above) covers the complement of Y; therefore X cannot be a subset of Y since F is an S-family.
Claim 4. Let X and Y be members of F with sizes different from m. Then XU Y :A [n]. Proof of Claim 4. In Claims 1-3 we may replace everywhere the family F by the family T. However, Claim 3 with F is equivalent to Claim 4 with F. The last two claims result in:
Claim 5.F - .gym is an SIC-family. Now we are in position to prove inequality (3.4). We use the corresponding inequality in Step 2 for B = SIC. We have
' E(fj+fn-j)+i Y (4+i)+_ j i
m-1
j=1
j=i+1
M-1
2iExai > 2 }. Obviously, F1 is a C-family (containing 0) and .T72 is an I V C-family (containing [n]). We have
P(a11 + ... +anAn < 12) _
pI XI (1 XE.
p)n-IXI
1
2) = E pIXI(1 -
p)n-IXI
XEF2
Let f; be the full profile of Xi (i.e., with coordinates fo and fn), and let w = (wo, ... , wn )T be given by wi := pi (l - p)n-', i = 0, ... , n. In order to obtain the lower bound (resp. the upper bound) we have to maximize wTf for f E it (%C) (resp. for f E µ(2(Iv-E)). For the lower bound, we take the "reversed" vectors from the table (extreme points) for B = I, add a 1 at a new
3.3 Boolean expressions
113
0-coordinate, a 0 at a new n-coordinate, yielding the vector
ci
(n)(n_l)(fl-_i)0 ' ...' 0)T
(n f ,
,
1 v(f') for all flows f' and c(S, T) < c(S', T') for all cuts (S', T'). We speak of integral flows f (resp. integral capacities c) if f, c : E -± N. Lemma 4.1.1.
(a) For any cut (S, T) and any flow f in N, we have f (S, T) - f (T, S) _
v(f) f (e) + I T f (e) - E f (e) a+=s
`e-ES-{s)
f(e)-57, f(e)
_ e-ES
e+ES
= f(S, S) + f(S, T) - (f(S, S) + f(T, S))
= f(S, T) - f(T, S)
c(S, T) - 0.
a+ES-(s)
4.1 Flow theorems
119
(b) From the proof of (a) we observe that v (f) = c(S, T) if f (S, T) = c(S, T) and f (T, S) = 0 - that is, if the conditions of the lemma are satisfied. But then for any other flow f' and cut (S', T') because of (a)
v(f') < c(S, T) = v(f) < c(S', T').
Remark 4.1.1. Note that we do not need the capacity constraints (4.2) to define
v(f ). Also without these constraints we have f (S, T) - f (T, S) = v(f ), since in the proof of (a) we needed for this equality only the conservation of flow (4.1). Now we are able to prove the central result of Ford and Fulkerson [183] and Elias, Feinstein, and Shannon [ 145], which we formulate first only for integral capacities since this case is interesting for combinatorial applications.
Theorem 4.1.1 (Max-Flow Min-Cut Theorem). Let N = (V, E, s, t, c) be a network with integral capacity. The maximum value of an integral flow in N equals the minimum capacity of a cut in N. Proof. By Lemma 4.1.1 we only have to show that there is an integral flow f and a cut (S, T) such that the condition of Lemma 4.1.1(b) is satisfied. We determine f recursively. We start with fo = 0 and assume that we have already constructed
integral flows fo, ... , f . Then we build recursi'--cly a vertex set Si which we consider later as a set of labeled vertices. We start with Sio Sio, ... , Sid are given. Let
{s} and assume that
Fib := {q c V - Sij : there is some p E Sid such that pq e E and 0}. c(pq) or qp E E and Case 1. Fib 0. Then we take some q E Fij and set Si,j+l := Si j U {q}; that is, we label vertex q.
Case 2. F,,j = 0; that is, for all p e Sij, q E V - Sid
.f(pq)=
c(pq)
if pq E E,
0
ifgpEE.
Since V is finite, Case 2 must hold for some j. At that moment or if earlier t E Sid we put Si := Si j. Now again two cases are possible: Case 2.1. t E S; . Then there must be a sequence of pairwise distinct points (po, pl, ... , pk) (called the flow-augmenting path) with an associated sequence
(el, ... , ek) of arcs such that po = s, pk = t and either et = pi-1 pi and f, (el) < c(ei) or
el = pipe-t
and
fi(ef) > 0
(ei is a forward arc) (el is a backward arc).
The flow-theoretic approach
120
Let
E+ := min{ c(ei) - f (ei) : el is a forward arc, 1 = 1, ... , k}, E- .= min{ fi (el) : el is a backward arc, 1 = 1, ... , k}, E
min{ E+, E-}.
Then we obtain the new flow fi+t by
f +1(e) :=
f(e) + E
if a is a forward arc,
f (e) - e
if e is a backward arc,
f (e)
if e is not an arc in the flow-augmenting path.
As illustrated in Figure 4.1, the flow conservation condition (4.1) is satisfied by
J. The capacity constraints (4.2) are satisfied by definition of E. Obviously, f +1 is an integral flow since E is an integer, and v (f +i) = v(f) + E. Note that E is positive, hence c > 1. t
t
t
t
Figure 4.1
Case 2.2. t 0 S; . Then f = f and (S, T) = (Si, V - Si) are a flow and a cut satisfying the conditions of Lemma 4.1.1(b). Since the value of any flow is bounded, for example, by the finite capacity of the cut c({s}, V - {s}) and since the flow value increases in each step by at least 1, after a finite number of steps Case 2.2 must appear. Remark 4.1.2.
(a) The Max-Flow Min-Cut Theorem remains true if there is not
given for any arc e a capacity constraint f (e) < c(e). For such arcs, we write c(e) := oc. But we must suppose that there is at least one cut of finite capacity (this was the only condition we needed at the end of the proof). (b) Instead of integral flows and capacities we can work with rational flows
and capacities. The result remains true since we can multiply by the common denominator of the capacities. From the practical point of view it is enough to consider rational capacities. If we have real capacities we do not have a better lower bound for c than 0. So
4.1 Flow theorems
121
we cannot show that the preceding procedure terminates. Moreover, there exist examples (cf. Lovasz and Plummer [356, p. 47]) where the flow values v(f) do not converge to the maximum flow value. But we still have some indeterminacy in our procedure. We have not specified which point q we label, that is, include into Si,j+l in Case 1 of our procedure. The possible points, the elare called fringe vertices. We collect iteratively the fringe verements of tices in a queue and take them by the principle "first in, first out" (FIFO). Tak-
ing the point q from the queue and putting it into the set Si,j+1, we obtain the following new fringe vertices, which we append at the end of the queue: jr E V - Si,j+l : qr E E and c(qr), orrq E E and fi(rq) > 0}. In order to find the flow-augmenting path in Case 2.1, we store in the queue together with the fringe vertices r the point q (with + or -) which led to r (by a forward, resp. backward, arc). Then the path can be determined recursively beginning from
t. Moreover, we store together with r the largest possible e = e(r) with which the flow change works. Obviously we have e(r) := min{e(q), c(qr) - f (qr)}, for a forward arc (resp. e(r) := min{E(q), f (rq)}, for a backward arc). In Case 2.1 we have then e := E(t). This is the so-called labeling algorithm using breadth first search (BFS). Without proof we mention the result of Edmonds and Karp [144]: Theorem 4.1.2. The labeling algorithm using BFS determines after O(I V I I E I2) steps the maximum flow and minimum cut in the network N = (V, E, q, s, c) with
c : E -+ R+. Since by this theorem the algorithm terminates (or because of a continuity argument) we have in particular:
Let N = (V, E, s, t, c) be a network with c : E -+ I[8+. Then max{v(f) : f is a flow in N} = min{c(S, T) : (S, T) is a cut in N}. Corollary 4.1.1.
The presented algorithm is easy to understand and to implement; see also Sedgewick [423] and Syslo, Deo, and Kowalik [449]. For more refined versions of flow algorithms, we refer to Tarjan [451], Jungnickel [279], and Mehlhorn [363]. Now let us assume that we have for N a further integral function a : E - * N, the cost function. The cost a (f) of the flow f is defined by a (.f)
.f (e)a (e), eEE
and we are interested in a flow having minimal cost and some value vo given in advance. The idea of Ford and Fulkerson [184] is to use a further function n : V -. N, the so-called potential function, for a primal-dual approach. Let us first prove an important lemma.
The flow-theoretic approach
122
Lemma 4.1.2. Let f be a flow of value vo in the network N with integral cost a. Then f has minimum cost with respect to all other flows of value vo if there is a function it : V -+ N with the following property:
f(pq) =
0
if
7(q) - ir(p) < a(pq),
c(pq)
if
n(q) - n(p) > a(pq)
for all pq E E,
(4.3)
and
(4.4)
7r (s) = 0.
Proof. For all pq E E, define a(pq) := a(pq) + 7r (p) - ir(q). Further let g be any flow of value vo. Then (using Lemma 4.1.1(a) and the conservation of flow)
g(e)a(e) - E 7r (p) E g(e) + E 7r (q) E g(e)
a(g) _ eEE
_
qEV
a+=q
ir(p)(g(V - (p), {p}) - g({p}, V - {p}))
g(e)a(e) + eEE
a-=p
pEV
PEV
_ E g(e)a(e) + rr(t)vo - 7r(s)vo eEE
_ 1: g(e)a(e) + von(t) eEE
Consequently,
a(g) - a(f)
J:(g(e) - f(e))a(e) > 0 eE E
since every item of the last sum is nonnegative: If a(pq) < 0, i.e., ir(q) - n(p) > a(pq) then
f(pq) = c(pq) > g(pq).
If a(pq) > 0, i.e., ir(q) - ir(p) < a(pq) then f(pq) = 0
< g(pq).
Remark 4.1.3. If f and it satisfy the conditions of Lemma 4.1.2, then we derive from the proof
a(f) = E c(e)a(e) + voJr(t). eEE:U(e) 0. As before, we come after finitely many steps (repeatedly augmenting the flow along paths) to a flow fk+l, a set Sk+1 of labeled vertices and a set Tk+l of unlabeled vertices with t E Tk+1 such that
c(e) fk4_l (e) =
I0
if7rk(e+) -7rk(e-) = a(e) and e- E Sk+l, e+ E Tk+1, if 7rk(e*) - 7rk(e-) = a(e) and e- E Tk+1, et E Sk+1 (4.5)
(the finiteness is clear if we have rational capacities, and for real capacities we can use the arguments of Edmonds and Karp in Theorem 4.1.2). We change in this phase the flow value from v(fk) to v (fk+l ). Note that we can also obtain every value between these two values since in Case 2.1 we need not change the flow by the maximum possible value E (we can take every c' with
0 < E' < E). Given fk+i, Sk+1, and Tk+1, we change the potential function defining 7rk(p)
nk+1(p)
l 7rk(p) + 1
if p E Sk+1,
if p E Tk+1
It is important to note that (4.3) and (4.4) remain valid in this case: (4.3) could be
violated (using the integrality of a and 7r) only if Irk (q) - Irk (p) = a (pq) and p E Sk+1, q E Tk+1 (but then (4.5) gives fk+l (e) = c(pq)) or if7rk(q) -7rk(p) =
a(pq) and p E Tk+1,q E Sk+1 (but then by (4.5), fk+1(e) = 0). Now we may continue with the labeling algorithm on the new network N,,+,. We show that if fk+l is not maximal in N, then we change the potential function only a finite number of times. Assume the contrary. Then we obviously have Sk+1 C Sk+2 c .... So there must be an index 1 such that S1 = S1+1 = .... But fk+1 is not maximal; hence by Lemma 4.1.1 there exists an edge e E E such that f (e) < c(e) and e- E S1, e+ E Tt or f (e) > 0 and e+ E Si, e- E T1. In view of (4.3), (4.4), and (4.5) (with l instead of k + 1), we must have 7ri(e+) - 7rl(e-) < a(e) (resp. 7rt (e+) -7r((e-) > a (e) ). So after a finite number of changes of the potential, the arc e is included into the network N,,,, for some m > 1, but then 5m+1 Q S1 U {e+} j St (resp. Sm+1 2 S1 U {e-} D St), a contradiction.
Summarizing, the algorithm works as follows: We change the flow a finite number of times, then the potential a finite number of times, then the flow a finite
The flow-theoretic approach
124
number of times, and so on such that (4.3) and (4.4) are satisfied. It is easy to see that there cannot be a flow-augmenting path in N,r from s tot if r (t) > E,e E a (e). By the preceding remarks we come to the maximal flow after at most >eEE a(e) changes of the potential function and then the algorithm terminates.
Remark 4.1.4. If for fixed potential 7r the flow value in the network N,r is increased by e, then the cost increases by 7r(t)E since the first sum in Remark 4.1.3 remains constant. Let us prove finally a technical lemma concerning flows in networks on directed acyclic graphs (dags). Here a dag is a digraph G = (V, E) in which there are no
points po,... , pk such that pi pt+1 E E,i = 0, ... , k - 1, and pk po E E. For example, the Hasse diagram of a poset is a dag. A directed elementary flow (def)
in a network N is a flow f for which there exist points s = po, pl, ..., (forming a directed chain) such that pipi+1 E E, i = 0, ... ,1 - 1, and
f (e) =
(1 0
pi = t
ife=pipi+l,i =0,...,1- 1, otherwise.
We define and denote the support of a flow in N by supp(f ):={e E E f (e) > 01. Lemma 4.1.3. Let N be a network on a dag with a directed chain from the source
to the sink, and let f be a flow in N. Then f is a nonnegative combination of directed elementary flows; that is, there are defs fl, ... , f and nonnegative real numbers ).1, . . . , Ai such that f = .1 ft + + Aifi. If f is integral, the 's can be taken to be integral, too.
Proof. We construct this combination by decreasing recursively I supp(f) 1. We may assume that I supp(f) I > 0. Then there is some arc pipi+1 such that
f (pipi+i) > 0. Suppose first that pi ¢ s and pi+1 # t. By the conservation of flow, (4.1), there must be arcs pi_1 pi and Pi+tPi+2 with positive flow. In the same manner we find arcs Pi_2pJ_1 and Pi+2Pi+3 and so on. No point can appear twice, because we have a network on a dag. Thus the procedure must end after finitely many steps, say at pops and pk_ 1 pk. Stop is possible only if p0 = s and pk = t. Now define ,1.1 := min{ f(pi pi+1), i = 0, ... , k- 1). Obviously, ,l1 > 0. Further set
fi (e) :_
r1 0
ife=pipi+1,i =0,...,k- 1, otherwise.
Evidently f := f - ,k l fl (e) is a flow on N with smaller support. Now we do the same for f', finding A2 and f2 and so on. We are done if supp(f) = 0. The claim concerning integrality is clear by construction.
4.2 The k-cutset problem
125
4.2. The k-cutset problem Although we are more interested in k-families, we start with k-cutsets (defined in this section) since the application of the flow method is a little bit easier for them, and of course we also need them later. For the poset P, let tt(P) be the set of all
maximal chains in P. A subset F of P is called a k-cutset if I F fl C I > k for all C E C(P). Let ko be the smallest number of elements in a maximal chain of P. Obviously, k-cutsets exist only for 0 < k < ko. The k-cutset problem is the following: Given a weighted poset (P, w), determine
ck(P, w) := min{w(F) : F is a k-cutset}. Here we set ck(P, w) := oo if k > ko. The k-cutset problem can be considered as an integer linear programming problem. If we relax it we arrive at the definition of a fractional k-cutset. This is a function x : P - III; such that
x(p) > k for all C E t(P), pEC
0 < x(p) < 1 for all p E P. In the fractional k-cutset problem we have to determine
ck(P, w) := min IpEP w(p)x(p)
x is a fractional k-cutset JJJ
Note that the fractional k-cutset problem reduces to the k-cutset problem if we require, in addition, that the solutions are integral. Here and in the following we deal simultaneously with the given problem and its dual. Given a general primal linear programming problem of the form
Ax+By+Cz < a Dx + Ey + Fz = b
Gx+Hy+Kz > c x>0 z < 0
d'x+eTy+fTz _*
max
where A, ... , K are matrices and a, ... , f are vectors of suitable dimension, its dual is defined to be the problem
ATU+DTV+GTw > d
BTU+ETV+HTw = e
The flow-theoretic approach
126
CTU+FTV+KTw < f
u>0 W z) := k
z(P) pEP
CEQ(P)
is maximized. By duality we have for any fractional k-cutset x (resp., in particular, for any k-cutset F) and any functions y, z satisfying (4.6) w(P)x(P) > Yk(Y, z)
(resp. w(F) ? Yk(Y, z)).
(4.7)
pEP
We can see this inequality also directly: We have Y(C) -
k
z(P)
pEP
CE(E(P)
(x(P)Y(C)) CEE(P)
pEC
1: ( 1:
z(P) pEP
x(P)Y(C) - z(P) I :SEX(P)W(P)-
/
PEP CEC(P):pEC
pEP
Theorem 4.2.1. Let ko be the smallest number of elements in a maximal chain of
the weighted poser (P, w). Then for each 0 < k < ko - 1 there exist a k-cutset Fk, a (k + 1)-cutset Fk+1 and functions y : tt(P) -> R+, z : P -* IR+ satisfying (4.6) such that w(Fk) = Yk(Y, z)
and
w(Fk+1) = Yk+1(y, z)
4.2 The k-cutset problem
127
Fk and Fk+1 are optimal k- (resp. (k + 1)-) cutsets and y, z are optimal solutions of the dual to the fractional k- and (k + 1)-cutset problem. In the case k = 0, we can choose z = 0. If w is integral then y and z can be chosen integral, too.
Proof. We will apply the algorithm of the proof of Theorem 4.1.3 in order to find Fk, Fk+1, y, and z. With our poset (P, w) we associate the network N = (V, E, s, t, c) with cost function a as follows: V := Is, t} U (p : p E P) U f p' p E P} (where s and t are new vertices), E := {sp : p is minimal in P} U { p't p is maximal in P} U {epl = pp' : p E P) U (ep2j = pp' : p E P) U { p1 p2 : PI < P2}. Here the union is understood in the multiset sense; we have two arcs from p to p', one denoted by epthe other by ep ). The capacity and cost are defined by
c(e)
a(e) :_
w(p)
if e = epl) for some p E P,
00
otherwise ,
if e = ep2) for some p E P,
1
otherwise .
10
The construction of the graph is presented in Figure 4.2 (the arcs are directed upward). Obviously, G = (V, E) is a dag. Moreover, with a directed elementary
t 5t 5
P S
G = (V, E) Figure 4.2
flow f along s, pi, p'1, P2, p'2, ... , pt, pt, t of value I we can associate the (max-
imal) chain C = (po < pl
0,
f, is a def for all i.
The flow-theoretic approach
128
Thus we may associate with f the function y : 1t(P) -* R+ defined by
if C corresponds to f as above, Y(C)
otherwise.
10 Note that in this correspondence
f(ep1)) + f(ep2)) =
Y(C),
(4.8)
CEt(P):pEC
a(f) = 1: f(ep2))
(4.9)
PEP
V(f) = E Y(C)
(4.10)
CEC(P)
Moreover, if it is a fixed potential function and n (t) < ko, then N, has a cut of finite capacity, because otherwise there would have to be a flow with arbitrary large value.
That is, there must be a def s, pl, p...... pi, p;, t going through ep,), ... , epr) and this is by the construction of N,r only possible if 7r(pi) - 7r(s) = 0, n (p') -
n(Pi) = 1, n(P2) - n(P'l) = 0,
.
,
ir(p') - n(Pt) = 1, n(t) - n(p) = 0
which implies 7r Q) - 7r (s) = ir(t) = 1 > ko = minimal number of elements in a maximal chain, a contradiction to it (t) < kp. Accordingly, the finiteness of our algorithm is ensured until ir(t) is changed from ko - 1 to ko. Now consider a situation where n is some actual potential with
ir(t) = k, 0 < k < ko, and f is a corresponding actual flow. Define F := {p E P : n(P') - 7r (P) = 1}. Claim 1. F is a k-cutset. Proof of Claim 1. During the algorithm, condition (4.3) is always satisfied;
hence (since c(ep2)) = oo) ir(p') - n(p) < 1 for all p E P and n(e+) ir(e-) < 0 for all arcs in E which are not of the form pp'. Consequently, if < ps) is a maximal chain in P then 7r(s) = 0, r(pl) C = (p1 < p2
p
e -=p
= w(P) - f(pp'),
(4.17)
f '(pq')
fp(e) = w(p) - .f (pp') - .f'(sp) + q i - that is, iff 7r (p) = i, 7r (p')
w(A1 fl Bi) = 0.
_
(4.26)
Assume the contrary. Then there is some p E Ai fl Bi with w(p) > 0 and 7r(p) > i, 7r (p') < i. The arcs sp and p't are saturated and the arc pp' has zero flow by (4.3). Thus there must be further points q, r E P such that f (pq') > 0
and f(rp') > 0 implying 7r(q') = n(p) > i and 7r(r) = it(p') < i. But by the construction of the network, q < p and p < r; that is, q < r and rq' E E. This is impossible since 7r(q') - 7r(r) > 0 and rq' has infinite capacity. From
4.3 The k-family problem
137
(4.23)-(4.26) we derive
v(f) = w(P) - w(F) + T f(pp'), pEF,
and summing up these equalities f o r i = 0, ... , k - 1 gives
kv(f) = kw(P) - w(F) + a(f), w(F) = k(w(P) - v(f)) + a(f ).
In the situation where the potential of t changes from k to k + 1 we get our desired k-families Fk and Fk+1. The additional assertions in the theorem follow as for k-cutsets. Corollary4.3.1.
We have dk(P, w) = dk (P, w) for all k.
Let us look also at the case of unweighted posets - that is, w - 1. Here Dilworth's Theorem 4.0.1 comes out very easily. Corollary 4.3.2.
We have
(a) (Greene and Kleitman [233]).
max{IFI : F is a k family in P}
= min
{min{ICI, k} : D is a chain partition of P CED
Moreover, the chain partition D can be chosen in such a way that the minimum in the above equation is attained not only for the parameter k but also for the
parameter k + 1. (b) (Dilworth [136]). The maximum size of an antichain equals the minimum number of chains in a chain partition of P. Proof. Of course, (b) follows from (a) with k = 1. So let us prove (a). If F is any k-family and D is any chain partition, then
IF n CI s
IFI = CED
min{ICI, k}. CED
Furthermore, let y and z be our optimal functions from Theorem 4.3.2. In the proof we noted that y and z satisfy (4.15) with equality. Since w - 1, y : (* (P) --+ {0, 11
and z : P -- {0, 1}. If y(C) = 1, then ICI > k, since otherwise we could change y(C) to zero and z(p), p E C, to 1 obtaining a smaller value in the objective function. The functions y, z provide the chain partition 0 whose chains are all
The flow-theoretic approach
138
C E V (P) with y(C) = 1 and the one-element chains {p} with z(p) = 1. Finally,
E min( ICI, k} = 6,(y, z) = max{BFI : F is a k-family), CE's
which proves the claim.
A partition 0 is called k-saturated if the minimum on the RHS of Corollary 4.3.2 is attained at 0. This corollary says that for each k there exist chain partitions that are simultaneously k- and (k + 1)-saturated. There are posets for which chain
partitions being k-saturated for every k do not exist. An example was given by Greene and Kleitman [233]; see Figure 4.6. Moreover West [466] constructed
Figure 4.6
posets having no chain partition that is k-saturated for any two nonconsecutive values of k (not exceeding the largest number of elements in a chain). Corollary 4.3.3.
The difference dk+1(P, w) - dk(P, w) is decreasing.
Proof. Again take the optimal families Fk, Fk+1 from Theorem 4.3.2 and consider
the flow f yielding these families. Then (note Claim 2 in the proof of Theorem 4.3.2)
dk+1(P, w) - dk(P, w) = w(Fk+1) - w(Fk)
= (k + 1)(w(P) - v(f)) +a(f) -(k(w(P) - v(f)) + a(f)) = w(P) - v(f). Since the flow value increases during the algorithm with increasing k, the difference
w(P) - v(f) decreases. We still continue the discussion of the proof of Theorem 4.3.2. An antichain partition 21 of P is defined similarly to a chain partition. Remember that every k-family is a union of k antichains. Corollary 4.3.4.
We have
(a) (Greene [229]) max{BFI : F is a union of k chains in P}
= min J - min( CAI, k} : 2l is an antichain partition of Pl AE2t
4.3 The k-family problem
139
Moreover, the antichain partition 2t can be chosen in such a way that for it the minimum in the preceding equation is attained not only for the parameter k but also for the parameter (k + 1). (b) (Mirsky [371]) The maximum size of a chain equals the minimum number of antichains in an antichain partition of P.
Proof. Again (b) can be derived from (a) with k = 1, but I encourage the reader to prove (b) directly.
For any union of k chains and any antichain partition 2t, we have IFI = EAE2l IF fl Al < F-AE21 min{IAI, k} since each antichain A can contain at most min{ I A 1, k} elements of F. Consequently, max < min. Let us look at the proof of Theorem 4.3.2 in order to prove max > min. Since w =_ 1, we are working with integral flows and integral functions y, z. For fixed potential n, let us increase the flow value always only by 1 (one could possibly do more along flow-augmenting paths, but in that case increase iteratively by 1). The maximum flow value in N is
w (P) = IPI . We achieve a situation where v (f) = I P I - k. Let n be the actual potential. Then by (4.19)
v(fp) = IPI - (IPI -k)=k. In view of (4.20) there are at most k chains C with y(C) > 0. Thus the union F of these chains is a union of k (possibly empty) chains. Because of (4.15) F contains at least all elements p with z(p) = 0, hence
IFI >- IPI -Ez(p)=IPI -a(f) pEP
(remember that z(p) = f(pp')). We find as follows the partition into antichains: Suppose that ko :_ 7r(t). In the proof of Claim 2 we worked with sets F, i = 0, ... , ko - 1. Each F is an antichain, since otherwise there would be p, q E F with p < q and ir(p) _ 7r(q) = i, 7r(p') = 7r(q') = i + 1, which is impossible since qp' is an arc of infinite capacity but having potential difference greater than its cost. Look at the antichain partition 2t whose antichains are FO, ..., Fkp_1 and the one-element classes { p} with p E P - F. Then, noting Claim 2,
min{IAI, k} < kko +IPI - IFI AE21
= kko+IPI -ko(IPI -v(f))-a(.f) = kko+IPI -kok-a(.f) = IPI -a(,f)
Thus we found F and 2t with
IFI > E min{IAI, k} AE21
yielding the desired inequality max > min.
The flow-theoretic approach
140
The situation where for fixed potential it the flow value f changes from I P I - (k + 1) to I PI - k yields the antichain partition 2t, which is optimal for the parameters k and k + 1. There exist several proofs of the results in Corollaries 4.3.2-4.3.4. Let us mention here Fomin [182], Hoffman and Schwartz, [270], Saks [405], and in particular Frank [186], whose proof for the unweighted case is the basis of our proof
of the weighted case. Generalizations and related results are by Edmonds and Giles [143], Linial [348], Berge [48], Hoffman [269], Saks [404], Gavril [219], Cameron [91], Cameron and Edmonds [90], Felsner [180], and Sarrafzadeh and Lou [414]. In particular for Dilworth's theorem there are a lot of different proofs, for example, by Dantzig and Hoffman [120], Fulkerson [205], Tverberg [454], and Harzheim [264]. For more informations, we refer to the above paper of Saks [404] and to the surveys of Hoffman [269], Schrijver [420], Berge [49], and West [465].
4.4. The variance problem Given a poset P, a function x : P -). 1[8 is called a representation of P if
x(q) - x(p) > 1 for all q > p. This notion was introduced by Alekseev [23] in order to find an asymptotic formula for the width of products of posets; see Section 7.2. An example of a representation
is the rank function r if P is ranked, and, in general, the height function h. Note that h can be calculated as a special case of the critical path method (cf. Sedgewick [423]): Label the minimal elements p of P with h (p) := 0. Let, after some steps, S be the set of labeled vertices. Then look in the next step for some minimal unlabeled point q (i.e., there is no unlabeled point below q) covering a labeled point p with largest label and put h (q) : = h (p) + 1. Continue until all points are labeled. By induction it follows easily that the calculated value h (p) is really the height of p. The height function of a small poset is illustrated in Figure 4.7. Now consider again positively weighted posets (P, w). For the representation x of (P, w) (i.e., x is a representation of P), we define the expected value (also called mean) µx and the variance ax by AX
w(P) E w(P)x(P) pEP
C1;11
W(P) 1 pEP
1
Y, w(P)(x(P) - µx)2 = w(P) E w(P)x2(P) pEP
-p,
4.4 The variance problem
141
h 3 2 1
0 Figure 4.7
respectively. Note that µx and ax can be defined for any function x : P -+ R. The variance problem is the following: Given a weighted poset (P, w), find a representation x such that aX is minimal with respect to all representations. Such a representation is then called optimal. Though it follows easily from a compactness argument that optimal representations really exist - in other words, that the minimum is attained - we will derive this fact from the finiteness of the algorithm below. The variance of an optimal representation is called the variance
of (P, w) and denoted by a2(P, w). First we need some technical details. The following lemma is obvious.
Lemma 4.4.1. Let x be a representation of (P, w), and let y(p) = x(p) + c for all p E P. Then also y is a representation of (P, w), and µy = µx + c,
2 =ax. 2
ay
For a nonempty subset F of P, we define its expected value by l,Lx(F)
w(F)
W(P)X(P), n
and put µx (0) := oo. Note that it, = µx (P). Lemma 4.4.2. Let x : P -+ R be any function and let F and I be disjoint subsets
of P with µx (F) < µx, and µx (I) > µx. Define yE : P -> R by
YE(P):=
x(p) + ew(I)
if p E F,
x(P) - ew(F)
if p E I,
x(p)
otherwise.
Then µyE = µx and a2 is strongly decreasing for 0 < e -< w(F)+w(I) (µx (I) -
Proof. The equality µyE = µx is easy. Moreover, if 0 < E
0 such that
g=Ao+ tAicoFi. Proof. We consider only increasing functions, the other case is analogous. It is enough to prove the statement for nonnegative functions g with Ao = 0 since each function can be written as a nonnegative function plus a constant, say A0. Now we prove this special case by induction on the support I supp(g) I. The case I supp(g)l = 0 is clear. If I supp(g)l > 0, put F1 := supp(g). Obviously, F1 is a filter. Furtherput A i := min{g(p) : p E Fl). Then g' = g-Ai pF, is nonnegative, increasing, and has smaller support size. The induction hypothesis applied to g' gives the result. Let E be the arc set of the Hasse diagram H(P) of P. For a given representation x, define
Ex:={eEE:x(e+)-x(e-)=1). The graph Gx = (P, Ex) is said to be the active graph. The poset whose Hasse diagram is Gx is called the active poset and denoted by Px. Finally, a function
f : P - IR+ such that
f (e) e+=p
f (e) = w (p) (x (p) - µx) for all p E P, e-=p
f (e) = 0
for all e E E- Ex
is called a representation flow on (P, w) relative to x. For the sake of brevity, we set
pf
.f(e)
of=p The following theorem was proven for unweighted posets by Alekseev [23]. We present our own proof of the weighted case.
4.4 The variance problem
143
Theorem 4.4.1. Let x be a representation of (P, w). Then the following conditions are equivalent:
(i) x is an optimal representation. (ii) >pE p w(p)g(p)(x(p) - µx) > O for all increasing functions g : Px _+ R.
(iii) ltx(F) > µx for allfilters Fin P. (iv) lµx (I) < Ax for all ideals I in P. (v) There exists a representation flow on (P, w) relative to x.
Proof. In view of Lemma 4.4.1 we may restrict ourselves to representations x with µx = 0. (ii) - (i). Let y be another representation of P with l y = 0. Then g = y - x is an increasing function from Px into R since fore E Ex we have y(e+) - y(e-) > 1 (y is a representation) but x(e+) - x(e-) = 1. We want to show that Qy > o ; that is, (4.27)
w(P)x2(P)
w(P)Y2(P) > pEP
PEP
From (ii) with µx = 0 and g = y - x we obtain
w(p)y(p)x(p) > 2
2
PEP
w(P)x2(P).
(4.28)
pEP
Evidently, (4.29)
w(P)(Y(P) - x(P))2 > 0. pEP
Addition of (4.28) and (4.29) gives the desired inequality (4.27).
(i) -* (iii). Let x be optimal and assume that there is some F in Px with gx(F) < 0 = /2g. Let I := P - F. Then obviously µ.x(I) > 0. Let := E1
1
w(I) + w(F)
min{x(e+) - x(e-) - 1
:
e c E - Ex}.
By definition of Ex, El > 0. Now consider the function yE from Lemma 4.4.2 for 0 < e < min{EO, E1). By the following reason, yE is a representation: Since x is a
representation we must consider only arcs pq with p E F, q E I. The set F is a filter in Px, thus pq E E - Ex. Finally
yc(q) - y, (p) = x(q) - x(P) - e(w(I) + w(F))
> x(q) - x(p) - min{x(e+) - x(e-) - 1 : e E E - Ex} > 1. From Lemma 4.4.2 we obtain oyE < ax, a contradiction.
The flow-theoretic approach
144
(iii) -> (ii). Write g in the form of Lemma 4.4.3. Then !
1 w(P) Ao + E.licPF I x(P)
Y" w(P)g(P)x(P) pEP
pEP
i=1
/
-+
Xi
i=1
(W(P)X(P)(P)) L pEP
!
_ EAiw(Fi)1tx(Fi) ? 0. i=1
(iii) H (iv). This assertion follows from the facts that F is a filter in Px if I = P - F is an ideal in Px and Ax (F) > gx iff Ax (P - F) < µx. (iii) H (v). For each representation x with µx = 0, let us define the network N = (VN, EN, s, t, cx), where VN := P U Is, t}(s and t are new vertices), EN
EU{sp: pEP}U {pt:pEP}(remember that E={pq: p 0 for every filter F in P. By the real version of the Max-Flow Min-Cut Theorem (Corollary 4.1.1) and in view of Lemma 4.1.1, the cuts ({s}, VN - {s}) and (VN - {t}, {t}) are minimal
if there exists a flow fN in N (the maximal flow) such that the arcs sp and pt (p E P) are saturated. Given such a flow fN, its restriction to E has the properties of a representation flow since for e E E
cx (sP) + p f = PJ + cx (Pt ),
pf
pf = max{0, w(p)x(p)} - max{0, -w(p)x(p)} = w(p)x(p)
and, of course, f (e) = 0 (since c(e) = 0) for all e E E - Ex. Conversely, given a function f satisfying the condition of a representation flow, we can extend f in a natural way to a flow fN in N such that the arcs sp and pt (p E P) are saturated. If we consider the variance problem as a quadratic optimization problem with linear constraints and convex objective function then the equivalence (i) H (v)
The flow-theoretic approach
146
follows from the theory of Kuhn and Tucker (cf. Martos [362, pp. 123 ff ]): The Lagrange function is L(x, f) := Q.,2 + f (e) (1 - x(e+) + x (e)), and the necessary and sufficient conditions for an optimal solution read:
x(e+) - x(e-) > 1 for all e c E,
P J - P f = w(P) w(P)(x(P) - I-Lx) for all p E P, f (e) ( 1 - x (e+) + x (e )) = 0 eE E
with f : E - R+. Of course, we may omit the factor 2 in the second condition, w so the arc-values of the representation flow can be interpreted as the Lagrange multipliers. We can use condition (iv) and the flow algorithm to construct an optimal representation together with the function f yielding then also o2(P, w). In order to do this we need some further condition: Let us say that (P, w) is in equilibrium with respect to the representation x if for every connected component C of the active graph Gx there holds ttx (C) = 0. Recall that a (connected) component in a directed graph is a maximal set C of vertices such that for any two elements v, w in C there exists a sequence
(v = v0, vl, ... , v = w) of vertices with the property that v, v,+l is an edge in the underlying undirected graph for all i E {0, ..., k - 1). The graph is called connected if it has only one connected component. Lemma 4.4.4. Let (P, w) be in equilibrium with respect to x. Then the values x(p), p E P, can be uniquely determined from the active graph.
Proof. The value of x on any point of any component C determines the values of x on all other points of C. The value of the first point must be chosen in such a way that the expected value of C becomes zero. Lemma 4.4.5.
Given a representation x, we can construct in a finite number of steps a representation y such that o > and (P, w) is in equilibrium with respect to y. Proof. Noting Lemma 4.4.1, we may assume that gx = 0. Let C1, ..., Ckx be the
components of G. Let ix`
{jE{l,...,kx}:gx(Cj) 0}; 1 := UjEJ,>Cj
4.4 The variance problem
147
We may assume that F 0, that is, I * 0 since otherwise (P, w) is already in equilibrium with respect to X. We put
w(I) + w(F)
E1
w(I)
min{x(e+) - x(e-) - 1
min{x(e+) -
x(e-) - 1
w
E4 :=
w(I) min{-/tx(Cj) :
E5 :=
e+ E I, e- E F),
e+ E R, e- E F),
:
rmn{x(e+) - x(e ) - 1
E3
:
:
e+ E I, e E R},
1
w(F)
min{µx(Cj) :
j E J,`1,
j E JX
and carry out the shifting yE defined in Lemma 4.4.2 with E := min{E1, E2, E3, E4,
65), which is obviously greater than 0. By checking all types of arcs we see that yE is indeed a representation. Moreover, E 2, be strict normal posets and let P
Qi, r(Q1) < ... < r(Qn)
(a) If the Whitney numbers of the posets Qi, i = 1, ..., n, are log concave, then P is [a,18)-normal and has (a, /3)-log-concave Whitney numbers, where
a := Ei=1 r(Qi), /3 := r(Qn) (b) If the Whitney numbers of the posets Qi, i = 1, ... , n, are strictly log concave, then P is strictly normal and has strictly log-concave Whitney numbers.
This corollary is helpful in proving the strict k-Sperner property:
Theorem 4.6.5. If P is an [a,18)-normal poset with (a, /B)-log-concave Whitney numbers, then P has the strict k-Sperner property for all k > max(1, /3 - a + 1 ). For these k, there are at most two maximum k -families.
Proof. Let us look first at the case k = 1 and a > /3 - that is, P is strictly normal and has strictly log-concave Whitney numbers. It is easy to see that there is some index h such that WO < WI < . . . < Wh > Wh+1 > > W". From the LYM-inequality (see Theorem 4.5.1), we derive for every antichain F IFI < "
Wh - n E i_o
I
F n Ni Wi
I
< 1,
and hence,
-
IFI < Wh,
where the first inequality is an equality if F C_ Nh and Wh > Wh+1 or F C Nh U Nh+l and Wh = Wh+1. In the first case we have only one maximum antichain, namely F = Nh. Assume that in the second case there is some maximum antichain different from Nh and Nh+1. Then 0 F n Nh # Nh, consequently
IFnNhI
IV(FnNh)I
Wh
Wh+1
but 0(F n Nh) and F n Nh+1 are disjoint since F is an antichain. Thus IFI = IFnNhI+IFnNh+11 < IV(FnNh)I+IFnNh+ll INh+1l,acontradiction to the maximality of F. Now let 1 < k < n and consider the chain Ck = (0 < 1 < < k - 1). Then Pk := P x Ck is strictly normal and has strictly log-concave Whitney numbers by Theorem 4.6.4. Let F be any maximum k-family in P. Then we construct the antichain A as in the proof of Theorem 4.3.1, which is by that theorem a maximum antichain in Pk. From the case k = 1 we know that we have at most two possibilities
for A, and A must be a complete level in Pk. Thus F is a union of k complete (neighboring) levels in P, and there are at most two possibilities for F.
4.6 Product theorems
175
Example 4.6.2. From Theorem 4.6.5, Corollary 4.6.2, and the remark after Corol-
lary 4.6.1, we obtain that the following posets have the strict k-Sperner property and that there are at most two maximum k families for all k = 1, 2. ...: B,,, L (q ) , A (q), projective space lattices, Q,, , F k, Int (S(k1, ... , k,)), modu+ k > k1 if k1 > ... > k, and k2 + lar geometric lattices and S(k1, ... , (the last result is due to Clements [110], other proofs can be found in Katerinochkina [289] and Griggs [240]). In [239] Griggs proved the Sperner property for the following poset PG : Suppose that [n] is partitioned into sets A 1 , ... , Am, and let I ; :_ [ai, bi ] , i = 1, ... , m, be nonempty intervals of numbers (we could also consider arithmetic progressions).
Let PG be the class of all subsets X of [n] with the property IX fl A; l E Ii, i = 1, . . . , m, and order these elements (i.e., subsets) by inclusion. Corollary 4.6.3. The Griggs poset PG is strictly normal and has strictly logconcave Whitney numbers.
Proof. Let Pi be the poset of all subsets Xi of A; with JXi I E Ii, ordered by inclusion. Obviously, Pi is isomorphic to the Ii-rank-selected subposet of the Boolean lattice BIA; I, which by the preceding remarks is strictly normal and has obviously strictly log-concave Whitney numbers. Moreover, PG is isomorphic to
P1 x
x P m where the isomorphism is given by X c [n ] H (X1 fl A 1, ... ,
Xfl Am). The statement follows from Corollary 4.6.2(b). Instead of partitions, West, Harper, and Daykin [467] took chains A 1 C A2 C C Am c_ [n] and defined similarly to the above poset PWHD of all subsets X of [n] with l X fl Ai I E Ii, i = 1, ... , m, ordered by inclusion.
Corollary 4.6.4. The West-Harper-Daykin poset PWHD is strictly normal and has strictly log-concave Whitney numbers.
Proof. Observe at first that we can assume, w.l.o.g., that for our intervals Ii = [ai, bi] the inequalities al < ... < am and bl < ... < bm hold and that Am = [n] (otherwise add [n] as a new element of the chain together with the last interval I m ). Let P k be the poset of all subsets X of Ak with I X f1 Ai I E Ii, i = 1, ... , k. Obviously, PWHD = Pm. We prove by induction on k that Pk is strictly normal
and has strictly log-concave Whitney numbers. The case k = 1 is easy. Now
consider the step 1 < k - 1 -+ k < m. Let Qk be the poset of all subsets of Ak - Ak_1, ordered by inclusion; that is, Qk = BIAkI-IAk-1I By induction and Corollary 4.6.2(b) Pk_1 X Qk is strictly normal and has strictly log-concave Whitney numbers. But Pk is that rank-selected subposet of Pk_1 X Qk whose members have cardinality in Ik; thus the result follows.
The flow-theoretic approach
176
The variance problem can be easier handled with respect to the direct product [152].
Theorem 4.6.6. Let x and y be optimal representations of (P, v) and (Q, w), respectively. Then the function z : P x Q - R defined by z (p, q) := x(p)+y(q) is an optimal representation of (P, v) x (Q, w). In particular, or 2((P, v) x (Q, w)) _
0`2(P, v) + a2(Q, w). Proof. Obviously, z is a representation of P x Q. Further it is easy to verify that µz = µx + Ay. Let f and g be representation flows on (P, v) relative to x and on (Q, w) relative to y, respectively, which exist by the supposition and by Theorem 4.4.1. Define the function h : E(P x Q) --). lR+ by
h((p, q)(p', q'))
w(q)f(pp')
if pp' E E(Px) and q = q',
v(p)g(qq')
if qq' E E(Qy) and p = p',
0
otherwise.
Then ((p, q)(p', q')) E E(P x Q) and z(p', q') - z(p, q) > 1 imply x(p') x(p) > 1 and q = q' or y(q') - y(q) > 1 and p = p'; that is, h((p, q)(p' q')) _ 0. Moreover,
(p, q)h - (p, q)h = w(q)pi + v(p)q+ - (w(q)pf + v(p)q9 ) g
= w(q)v(p)(x(p) - µx) + v(p)w(q)(y(q) - µy) = (v x w)(p, q)(z(p, q) - µz);
that is, h is a representation flow relative to z, and by Theorem 4.4.1 z is an optimal representation. Finally,
Q2((P, v) x (Q, w)) 1
(v x w)(P x Q) 1
(v x w)(p, 9)z2(p, q) - µz (p,q)EPxQ
v(P)w(Q) peP,gEQ
v(p)w(q)(x(p) + y(q))2 - (µx + µy)2
= v2(P, v) +Q2(Q, w), and the proof is complete.
We have seen that the direct product of normal posets is normal under some suppositions on the Whitney numbers. One cannot totally omit these suppositions. The product of the two unweighted normal posets in Figure 4.10 does not have
177
4.6 Product theorems
x Figure 4.10
the Sperner property; that is, it is not normal. Because of Theorem 4.6.6 the direct product of rank-compressed posets is rank compressed. The situation is different if we consider rankwise direct products (this result is essentially due to Sali [409]).
Theorem 4.6.7. If (P, v) and (Q, w) are normal posets of same rank, then (P, V) Xr (Q, w) is normal, too.
Proof. Let f : Ei (P) -+ R+ and gi
: E; (Q) - 118+ be functions satisfying the conditions of Corollary 4.5.1, correspondingly. Define h1((p, q)(p'q'))
Ei(PXrQ)-SIR+,i=0,...,r(P)-1, by hi ((p, q)(p', q')) := f (pp')gi (qq'),
pp' E Ei (P),
qq' E Ei (Q)
Then
v(p)
(p, q) h; _ p' >p, q'>q
w(q)
f (Pp')gi(gg') = pf.ggi = v(Ni(P)) w(Ni(Q)) _
(V Xr w)(p,q)
(V Xr w)(N,(P Xr Q))'
and the corresponding equality for (p, q)h holds, too. By Corollary 4.5.1, (P, V) Xr (Q, w) is normal. More generally, using Theorem 4.6.3 and straightforward calculations, we obtain that the rankwise direct product preserves I-normality and I-log concavity.
Example 4.6.3. The poset of subcubes of a cube SQ(k, n) and the poset of square
submatrices of a square matrix SM(k, n) are strictly normal with strictly log concave Whitney numbers.
However, the rank-compression property is not preserved by rankwise direct product that can be seen by the unweighted rank-compressed poset given in Figure 4.11 (see [59]): The indicated elements form a filter F. It is easy to see that F xr F in
178
The flow-theoretic approach
P x, P does not satisfy the inequality (iii) in Theorem 4.4.1; that is, P x, P is not rank compressed.
Figure 4.11
5
Matchings, symmetric chain orders, and the partition lattice
This chapter should be considered as a link between the preceding flow-theoretic (resp. linear programming) approach and the succeeding algebraic approach to Sperner-type problems. The main ideas are purely combinatorial. The powerful method of decomposing a poset into symmetric chains not only provides solutions of several problems, it is also very helpful for the insight into the algebraic machinery in Chapter 6. In particular, certain strings of basis vectors of a corresponding vector space will behave like (symmetric) chains.
5.1. Definitions, main properties, and examples Throughout let P be a ranked poset of rank n := r(P). We say that the level Ni can be matched into the level Ni+i (resp. Ni_1) if there is a matching of size Wi in the Hasse diagram Gi = (Pi, Ei) (resp. G;_1 = (P1_1, Ei_1)) of the {i, i + 1}(resp. {i - 1, i }-) rank-selected subposet (recall that a matching is a set of pairwise nonadjacent edges or arcs).
Theorem 5.1.1. If there is an index h such that Ni can be matched into Ni+1 for 0 < i < h and Ni can be matched into N!_1 for h < i < n, then P has the Sperner property. Proof. If we join the arcs of the corresponding matchings at all points that are both starting point and endpoint, we obtain a partition of P into saturated chains (isolated points are considered as 1-element chains). Each such chain has a common point with Nh. Consequently, we have Wh chains and Dilworth's Theorem 4.0.1 implies
d(P) < Wh, that is, d(P) = Wh. There exist several criteria for the existence of such matchings. 179
Symmetric chain orders
180
Theorem 5.1.2 (Hall's Theorem [251]). Let P be a poset of rank n = 1. Then No can be matched into N1 i,/fforall A C No there holds Al I< IV(A)I. Proof. The inequality I A I < I V (A) I for every A C No is clearly necessary, so let us look at the sufficiency. Let S be a maximum antichain in P, A := No n S, B :_ N1 fl S. Then V(A) fl B = 0; that is, ISI Al + IBI IV(A)I+IBI W1.
Consequently, d(P) = W1. By Dilworth's Theorem 4.0.1, P can be partitioned into W1 chains. Obviously, every chain must contain exactly one element of Ni. The 2-element chains form the desired matching.
Theorem 5.1.3 (R. Canfield [96]). Let P be a poset of rank n = 1 without isolated minimal points. If for all p, q E P with p < q there holds I V (P) 12: I A (q) then No can be matched into N1.
Proof. We consider the network N = (V, E, s, t, c) where V := No U Ni U {s, t}, E := (sp : p E No) U {pq : p < q} U {qt : q E Ni} and c(sp) := 1 for
pENo,c(pq):=ooforp 20 .
Symmetric chain orders
202
We conclude that a2
IV(n)I >
(5.5)
20.
Now consider any or E N,+1. It has n - i - 1 < n - i blocks. We find the predecessors of a in Q; by splitting a block of size 2b or less under additional conditions. Consequently,
IA(or)I < (n - i)22b = (n -
i)n1-813.
(5.6)
Let n be sufficiently large such that 826 > 20n-3/3. Then we obtain from (5.5) and (5.6) by straightforward computation IV(n)1 >_ Io(a)I
and by Theorem 5.1.3 the level Ni can be matched into Ni+1 in our poset Q, that is, also in II,,. (b) Since there are 21 BI-1-1 possibilities to partition a block B into two blocks,
we have, for r E Ni and with k := n - i = number of blocks in n,
IA(n)I =
(21B1-1
_ 1) >
k(2n1k-1
_ 1)
B block in n
where the last inequality follows by applying Jensen's inequality to the convex
function f(x) = 2x-1 - 1. We have k > 11 g2, which implies 21'k > n and further
IA(n)I>k(2-1). Moreover, for a E N1_1 obviously,
IV(Q)I = \k
1).
We have (k21) < k(2 - 1) if k < n - 3, and2this is true for n > 6. Thus in this case I 0 (n) I > I V (or) I, and Theorem 5.1.3 (in its dual form) gives the matchings. For n = 5, the assertion of the theorem is easily checked by inspection.
It is very interesting that the bounds in Theorem 5.4.4 are the best possible. Theorem 5.4.5 (Canfceld [96]). For every 3 > 0, there exists an no (8) such that for all n > no (8) there holds in the partition lattice II,,:
(a) The level N1 cannot be matched into Nr+1 if i > n - (1 - 8) jl (b) The level Ni cannot be matched into Ni_1 if i < n - (1 + 3) i1
We will prove only part (a) by a construction that is partly due to Shearer [428]. The proof of part (b) relies on the application of a local limit theorem (together
5.4 Semisymmetric chain orders
203
with an estimation of the remainder). Since we will not use the result of part (b) further in this book, the interested reader is referred to [96], but note that in Chapter 7 we will discuss applications of local limit theorems. Proof of (a). Let, w.l.o.g., S < 1 and let i > n - (1- 8) log n . By Theorem 5.1.2 it is enough to find some A c Ni such that I V (A) I < I A 1.i Let log n
(1-8)log4 log 4 < n Every element of Ni has n - i < n (1-a) togn - m blocks. Let A be the set of those 1-g/2 partitions from Ni at least n 1-s/2 of whose block sizes equal 2m, at least n of whose block sizes equal or exceed 3m, and all of whose block sizes belong to the set M:= {m, 2m, 3m, 3m + 1, 3m + 2, ...}. Since we have for every fixed c > 0 and sufficiently large n that
n 1-s/2cm
clog n
IV(A)l2(2m)n(5.7)
Using the fact that (2m) is the largest of 2m + 1 binomial coefficients whose sum is 22m we obtain 2m
>
22m >
1
2m+1
M
1
- 2m+1
22(1(
7) lo "ng
-1
=
1
n1/(1-s)
4(2m+1)
n>
Consequently, by straightforward computation, 1
2m
2m
nt-s/2 >
1
8(2m + 1)
1 (n)2 2 \m
if n is sufficiently large. Now from (5.7) and (5.8) we derive I A I > I V (A) I .
(5.8)
Symmetric chain orders
204
In addition to this result, Canfield [96] proved, using some earlier results of Mullin [376], that the transition from the existence to the nonexistence of matchings between neighboring levels is not chaotic but abrupt - that is, there are sequences
Ln and Rn such that Ni can be matched (resp. cannot be matched) into Ni+1 if i < Ln (resp. i > Ln), and analogously for Rn and Ni -> N1_1. Moreover, both Ln and Rn grow by at most 1 when n is increased by 1. We may use this result to show that the partition lattice rIn does not have the Sperner property if n is large enough. Before doing so, we need a localization of the largest Whitney number of rln, or, equivalently, of the maximum Stirling number of the second kind. The sequence of Stirling numbers {Sn,k : k = 1, ... , n} is strictly
Lemma 5.4.1. log concave.
Proof. We proceed by induction on n and use the basic recurrence Sn,k = kSn -1,k+
Sn_1,k_1 (applied to Sn,k, Sn,k_1, and Sn,k+1). The cases n = 1, 2 are trivial. Concerning the induction step we have (for 2 < k < n - 1) 2
2
Sn,k - Sn,k-1Sn,k+1 = (Sn-1,k-1 - Sn-1,k-2Sn-1,k)
+(k2Sn_l,k
- (k2 -
1)Sn-l,k-1Sn-1,k+1)
+ (k + 1)(Sn-l,k-I Sn-l,k - Sn-1,k-2Sn-1,k+1) The first two summands are positive by induction. Moreover, by induction, 2
2
Sn-1,k-1Sn-l,k >- (Sn-l,k-2Sn-l,k)(Sn-1,k+lSn-l,k-1), which implies that the third summand is also nonnegative.
Note that this result shows that l1n is rank unimodal. Let kn be that number
for which Sn,I < Sn,2
Sn,n;
that is, Wn_k is the largest Whitney number of fIn. Canfield [94] determined kn precisely for sufficiently large n. Almost the same formula was obtained by Jichang and Kleitman [277]: Let x be the root of (x + 2) log x = n - 2. Then kn E 1 LX + 12J, [x + 12J + 1). Moreover, kn = Lx J + 1 in "almost all cases." The proof is very complicated. From a recent result of Benoumhani [45] on the maximum coefficient of a polynomial whose roots are all negative real, it follows
that (for all n) Ikn - (n - A,(rln))I < 1 where lr(fn) is the expected value of the rank function of 11n. With (6.50) this implies Ikn - BBn' + 11 < 1. Here we need only a slightly weaker result: Theorem 5.4.6 (Harper [256]). We have kn - T-n as n -> oo. o
5.4 Semisymmetric chain orders
205
Proof. We present a short proof that is essentially due to Rennie and Dobson [395].
It is enough to show that for every e > 0 we have
(1 - e)logn + 0(1) < kn < (1 +)logn
+
0(1).
(5.9)
Claim 1. If 1 < k < n, then kn-k Sn,k < Proof of Claim 1. Let us construct partitions with k blocks by putting n balls labeled with the numbers 1, ... , n into k indistinguishable boxes such that each box obtains at least one ball. We put the balls with numbers 1, . . . , k into the boxes such that in each box there is exactly one ball. For the remaining (n - k) balls, we have kn-k possibilities for the distribution. This yields kn-k pairwise different partitions. Thus the first inequality is proved. However, every partition can be constructed in the following way. Put ball number 1 into one box. Choose k - 1 other balls (in (k_i) different ways) and put them into the remaining boxes such that in each box there is exactly one ball. Distribute the remaining n - k balls as before. This yields the upper bound (n-I)kn-k
Claim 2. If 1 < k < n, then (k) < k- tn nn
Proof of Claim 2. There are nn possibilities to put n balls labeled with 1, ... , n
into n boxes labeled with 1, . . . , n. Some of them can be obtained as follows: Choose k balls ((k) possibilities). Put them into the boxes 1 , ... , k (kk possibilities). Put the remaining n - k balls into the boxes k + 1, ... , n ((n - k)n-k possibilities). Clearly we obtain pairwise different distributions. Consequently, (k)kk(n k)n-k < nn, and Claim 2 is proved.
-
Let
kc := k*
.
n
clog n + 0(1), long
n
(c is a positive constant),
+ 0(1)
(we have the term 0(1) to ensure that k and k* are integers). Claim 3. We have Sn,kc < Sn,k* if c 1 and n is sufficiently large. Proof of Claim 3. We write briefly k := ke. In view of Claims 1 and 2 (together with the trivial inequality (k-i) < (k)), it is enough to show that nn
kn- k < k*n-k
kk(n - k)n-k which is equivalent to
n log n + (n - 2k) log k - (n - k) log(n - k) < (n - k*) log k*.
Symmetric chain orders
206
If we subtract the LHS from the RHS and use log(l + x) = x + o(x) we obtain the term n
1-
1
(logn -log logn +o(1)) -n logn
to
-n I 1 - t cgn) (log c + log n - log log n + o (1)) o \\\/
+n I 1 - logn) (logn +o(1))
=n(c-logc- 1+0(1)). From Proposition 2.6.1(b) it follows that for sufficiently large n the last term is positive if c 0 1, and this proves the statement in the claim. From Claim 3 we derive for E > 0 that k1_6 < k* < kl+E which is our desired inequality (5.9). As already mentioned we are now able to prove Canfield's result, which was a markstone in the development of the Sperner theory. It answers a question of Rota [402] negatively. Simplifications of the first complicated proof have been given by Shearer [428] and Jichang and Kleitman [277]. Theorem 5.4.7 (Canfield [93]). For sufficiently large n, the partition lattice Iln does not have the Sperner property.
Proof. Take some small 8 > 0 and let n be large enough such that (1 - 8) x log 4 0 > kn + 1 (apply Theorem 5.4.6). Let i := n - kn - 1 (i.e., W;+1 is the largest Whitney number of rln) and take the set A C_ Ni from the proof of Theorem 5.4.5(a). We have seen that for sufficiently large n, I A I > I V (A) I. Obviously, the
set S := A U (N;+1 - V(A)) is a Spemer family whose size is greater than W,+1.
Theorem 5.4.7 and Corollary 4.5.3 imply that IIn cannot be normal if n is sufficiently large. There is a sharper result:
Theorem 5.4.8 (Spencer [435]). Thepartition lattice nn is not normal ifn > 20.
Proof. First let n be even. Let A be the set of those partitions from Nn-2 both of whose block sizes equal Then A(A) consists of those partitions from Nn-3 that have one block of size 2. We have 2.
Wn-2 = Sn,2 = 1(2n - 2), Wn-3 = Sn,3 = 3 (3n - 3 . 2n + 3),
5.4 Semisymmetric chain orders
IAI = 2
207
(nn),
z
Ii (A)I =
()Sn,2. 2
In a straightforward way one obtains I
A
I
Wn-2
I
0 ( A )I
1ff3 n-1
wn-3
20. For n odd, we may use the same arguments if we work with the set A of partitions from Nn-3 that have {n } as one block and whose other block sizes both equal '2 11.
With the partition lattice one can also associate the poset Pin of unordered partitions of an integer, ordered by partitioning the items. More precisely, Pin
(a = (al, ... , an) E N : E"=1 io; = n}, and we have o < r if there are i, j E [n] such that ai+j = ri+j - 1 and a; = ti + 1, aj = rj + 1 if i # j (resp. a; = r; + 2 if i = j). The symmetric group Sn on [n] induces in an obvious way a group G of automorphisms of Fln. It is easy to see that the quotient IIn/G equals Pin. Moreover, if we define w : Pin - R+ by n!
!
n
a;!(i!)O'i
then (Pin, w) is the weighted quotient of (IIn, 1), and it does not have the Sperner property for large enough n by Theorem 4.5.6 and Theorem 5.4.7. However, the
following problem is still open: Does the unweighted poset Pin of unordered partitions of an integer have the Sperner property?
6
Algebraic methods in Sperner theory
In this chapter we are concerned with unweighted, ranked posets only. Algebraic characterizations for the existence of certain matchings in graphs have been known for many years (cf. Lovasz and Plummer [356], and note in particular the results of Kasteleyn [286], Perfect [380], and Edmonds [1421). We have seen that chains in posets can be constructed by joining matchings between consecutive levels. By Dilworth's theorem, the size of chain partitions is related to the width of a poset.
For these (and other related) reasons, it is worthwhile to develop an algebraic machinery that answers many questions in Sperner theory. An essential step in this direction was undertaken by Stanley [439], who used deeper results from algebraic geometry (the Hard Lefschetz Theorem) in order to prove the Sperner property of several posets. Further work (in particular that of Proctor [388]) allows us to stay on a more or less elementary linear algebraic level. It is interesting that Erdo"s-Ko-Rado type theorems can also be proved using this approach. Important contributions were those of Lovasz [355] and Schrijver [419] and some culmination was achieved by Wilson's [470] exact bound for the Erdo"s-Ko-Rado Theorem in the Boolean lattice. Besides this result, Wilson strongly influenced, in general, the development of the algebraic extremal set theory and, together with Kung (e.g., [327, 328]), the theory of geometric (and related) lattices. Kung's main tool is the finite Radon transform discovered by E. Bolker (see [328]), but we will formulate the results in the language of injective linear operators. Extremal set problems are discussed in Section 2.5. In this chapter (Section 6.5) we apply the fundamental observation that the image of a subspace of a vector space by a linear mapping has dimension not greater than the dimension of the preimage. For further information, the reader is referred to the book of Babai and Franki [35]. Though the theory is easier to present under the supposition of rank symme-
try we will consider first the general case and later specify the results to ranksymmetric posets. An important tool is the Jordan decomposition of nilpotent 208
209
6.1 The full rank property and Jordan functions
maps considered by Saks [406] and Gansner [213]. We introduce here the Jordan function in order to derive many results by algebraic manipulations only. We prefer to work throughout this chapter with linear operators instead of (incidence) matrices.
6.1. The full rank property and Jordan functions Throughout, let P be a ranked poset with rank function r of rank n := r(P). With P we associate the poset space P, which is the vector space of all functions cp
: P - R with the usual vector space operations. With the element p of P we
associate its characteristic function cop E P defined by
(1 (pp(q)
0
ifq=p, otherwise.
For the sake of brevity, we write p instead of cpp, but 0 denotes the zero vector. Obviously, [p : p E P} is a basis of P. Thus every element cp of P has the form cp = F_pE p A p p, where g p is a real number, and P can be considered as the vector
space of all formal linear combinations of elements of P with real coefficients. With the standard scalar product
CE APP E pEP
PEP V Pj
:= pEP f'LPV P
the space P becomes a Euclidean space (here we are working only with the field of real numbers). With a subset F of P we associate the subspace F of P, which
is generated by {: p E F). Obviously, the dimension of F, denoted by dim F, equals IFI._Note that P = No ® ® N,,, where ® denotes the direct sum. Any basis B of P with the property B = U"=oB fl N1, that is, all basis elements belong to some N1, is said to be a ranked basis. In particular, {p : p E P} is a ranked basis. In the following we consider linear operators P P. We define VP to be the operator for which 4)IP (cp) = 4)(41(cp)) for all cp E P. Further let (D ' :_ 4) ... 4) (i times) if i > 1. It is useful to define (Di to be the identity operator I if j < 0. For a subspace E of P, let 4'(E) := {cp(cp) : co E E} and 4)IE be the restriction
of 4) to E. Let V be the adjoint operator of 4), that is, 4)*
P - P with
(4)(cp), fl = (cp, (D*(,lf)) for all cp, f E P.
If j 0 (0, ... , n), let Nj = {0}. A linear operator V P_ -> P (resp. A P_ -- P) is called a raising (resp. lowering) operator if V (N,) c_ N;+1 (resp. L (N;) c N, _ 1 for all i). A raising (resp. lowering) operator 0 (resp. i) defined on the basis { p : p E P} by V(P)
/
1: c(p, q)q I resp. O(q) q:q>p
\
d(p, q)P, f p:p h, BSI < IA(S)I. By Theorems 5.1.1 and 5.1.2 this is sufficient for the proof. Indeed, if i < h, W, < W,+1 implying that V,,1+1 is injective. Since V is order raising, V,,1+1 (S) is a subspace of V(S). Hence
ISO =dims= dimV1,,+1(S) :!S dim V(S) = IV(S)I. If i > h, we may argue analogously using V*1,1. For a product of two chains, there is a nice way to compute the rank of Lefschetz matrices:
q'} 1,
Algebraic methods in Sperner theory
228
so (b) is proved.
From our claim it follows for k > 0 that
fvk = OLf and gV*k = ALg. Let 0 < i < j < r(P). Then we have in particular
fV, j = VLIJ f and gV = ALJ,g. If W;(P) < Wj(P), then OL;j is injective by supposition and Lemma 6.1.2(a). Since also f isinjective, 0j j must be injective, too - that is, of full rank. If W, > Wj then L.Lj; is injective by supposition and Lemma 6.1.1(b) and Lemma 6.1.2(b). Since g is injective, V must be injective; that is, by Lemma 6.1.2(b), Vi j is of full rank.
Applications of this theorem will be presented in the next section. We conclude this section with a proposition we will need in Section 6.4.
Proposition 6.1.1. Let P be a ranked poset for which there exists a group G of
automorphisms of P whose orbits are the levels of P, and let 0 < i < j < r(P). Ifrank ;j(VL) < Wj, then p c OL;j (N;) for all p E Nj. Proof. Each element g of G defines a linear mapping yg : P --- P by yg(p) g(p) for all p E P. We have
Yg'L = VLYgfor all gE G since for any p E P
g(q) _
Yg'L(P) q:q>p
q = OLYg(p) q:q>g(p)
Consequently,
YgVL;j (0 = VL;;Yg(O for all lp E Ni.
(6.12)
Assume that for some p E Nj there is a tp E N; such that VL, (rp) = p. Then for each q E Nj, there exists by our supposition an element g of G such that q = g(p). From (6.12) we obtain q = yg(p) = YgVL,; (cp); that is, q E OL;j (N1). Thus Nj = VL11(N;), which contradicts Wj > rank; j(OL). This proposition can be applied to B,, and L,, (q) (take the automorphisms generated by permutations of the elements (resp. by bijective linear transformations)).
229
6.2 Peck posets
6.2. Peck posets and the commutation relation In the preceding section we learned that in order to prove the strong Sperner property it is enough to find an order-raising (resp. order-lowering) operator with the full rank property. A first approach is given in Theorem 6.1.9. In this section we mainly restrict ourselves to rank-symmetric posets, for which it is easier to find such operators with the full rank property. In honor of the "dummy" mathemati-
cian G.W. Peck and his best friends Graham, West, Purdy, Erd6s, Chung, and Kleitman, a ranked poset P is called a Peck poset if it is rank symmetric, rank unimodal, and if it has the strong Sperner property. One finds equivalent conditions in Theorem 6.1.6 where one has always to add rank symmetry. In particular, P is Peck if it is rank symmetric and there exists an order-raising operator 0 with the full rank property. If, in particular, the Lefschetz raising operator has the full rank property, then we speak (in the case of rank symmetry) of a unitary Peck poset.
Lemma 6.2.1. Let P be a ranked poset of rank n. It is Peck iff there exists an orderraising operator 0 such that for the Jordan function there holds J(V , P; x, y) =
F(P; x)yn where F(P; x) is the rank-generating function of P. The poset P is unitary Peck if'J(OL, P; x, y) = F(P; x)yn. Proof. Let P be Peck. Then we find a V with the full rank property. We have
sij =0ifi+j 0n,i < j. We verify this fori+j < n, the case i + j > n is analogous. By Theorem 6.1.3 and Lemma 6.1.6, and in view of the supposition,
0 > Wi - Wj+i = ranki,i - ranki,j+i =
T suv>_ sij u t.
(6.14)
Obviously, Ei,t C Ni for all i > t. Lemma 6.2.5.
(lto, ...,
Suppose that (0, t,) has in P property C with the sequence Let i, j, t be natural numbers with i > t and let aj,g,h be defined
as in Lemma 6.2.3.
(a) All nonzero elements of Ei,t are eigenvectors of Aj Vj I N to the eigenvalue a'j, j+i-t,t(b) All nonzero elements of Ei,t are eigenvectors of;MKijj to the eigenvalue
aj,i-t,t (c) If, in addition, V = A* then Ei,t, is orthogonal to E1,t2 for all 0 < t1 < t2 < i.
Proof. For (a) and (b), let cp E Ei,t, that is, cp = Di _t (cp'), where cp' E Nt , 0 ((p') _ 0. By Lemma 6.2.3, &jVj+i-t((p')
= a,Gj+t-t,roi-t((P') = aj,j+t-ta(co), (b) OJOJ(tP) = VJEJ0i-t(tP) _ VJaj,i-t,tOi-t-J((P') aj,i-taco. (a) OJOf(gp) =
(c) Let cpi E E1, t,, that is, cp E Di-t!(coi) where tpi E Nt,, O(cpl) = 0, I = 1, 2. We have
41, tp2) = (0`-t' (w1), o` t2((P2)) = (cPi ` t`O`-`2((P2)) = (cPi, 0) = 0
since i - t1 > i - t2 and O(cp'2) = 0 (use again Lemma 6.2.3). Theorem 6.2.5. Under the supposition that (V, O) has in P property C with the regular sequence (It.o, ... , we have
(a) Ni = E1,o ® ... ® Ei,, (direct sum) for all 0 < i < 2 (b) & VJ I is injective for all 0 < i < 2 and 0 < j < n - 2i. Proof. In (b) we suppose j > 0 since the case j = 0 is trivial. We proceed by induction on i_= 0, ... , L 2 J and prove (a) and (b) simultaneously. _Let i = 0. Then Eo,o = No, and by Lemma 6.2.5(a), Eo,o is the eigenspace of AjVf INo to the eigenvalue aj,j,o, which is nonzero for all 0 < j < n by Lemma 6.2.4 (resp.
(in the case j = 0) by definition). Thus AJVJN0 is injective for all 0 < j < n. Now consider the step < i -* i (i < 2). From (6.5) (replace 0 by 0) we may derive
dimkert,t_1(i) = Wt -rankt,t_t(0) > Wt - Wt- 1,
t = 0, ...,n.
Thus
dim Ei,i = dim keri,i_ i (0) > Wi - WJ_1.
(6.15)
Algebraic methods in Sperner theory
236
By the induction hypothesis, A"-2tVn-2tIN is injective for all 0 < t < i. Since is injective. By the t < i < 2 there holds i - t < n - 2t; thus also definition of E1,t (see (6.14)) we have
dim Ei,t = dim kert,t_1(A) > Wt - Wt-1,
0 < t < i.
(6.16)
From (6.15) and (6.16) we derive i
dim E1,t > Wi = dim Ni. t=o
+ E1,i is direct (note Thus for (a) it is sufficient to prove that the sum Ei,o + + Ei,1 c N1). Let cot E E1,t, t = 0, ..., i. We have to show that that Ei,o + + (pi = 0 implies (pt = 0 for every t = 0, ... , i. Let (pt = Di-`(cpt), tpo + + Vi = 0 is equivalent where c p i E Kt, A(tp') = 0, t = 0, ... , i. Hence tpo + with 0,
o + ... + Do(tpi-1)) + toj = 0. Applying 0 to both sides we obtain
00 (cp'o' + ... + (pl' 1) = 0, where tpt = pi -r(cpt) E E1_ 1,t, 0 < t < i - 1. By the induction hypothesis AVON. , is injective; consequently, coo
{
...+toi
1 =01
and since by the induction hypothesis Ni_1 = Ei_l,o® ... ® E1_1,1_1, it follows that tot' = 0 implying (since cpt = V (W' )) that tot = 0, t = 0, ... , i - 1. But then clearly also coi = 0. (see For (b), it is sufficient to look at the eigenvalues aj,j+i_t,t of A& VJ Lemma 6.2.5(a) and use the proved part (a)). In our case 0 < i < 2 and 0 < j
d+ (U) iff pn-d > pe iffr(U) < n, d-(U) < d+ (U) iff pn-d < pe iff r(U) > n; thus, the degree-property holds.
It is conjectured (see [443]) that the subgroup lattice of Gk(p) has the Sperner property for all positive integers k, but already the case k = 3 is open. On the other
hand, Butler [88] proved the rank symmetry and rank unimodality of Ga(p) for every A.
Let us now turn to geometric lattices. We already know that they do not, in general, have the Sperner property. But there is a nice conjecture of Kung [329]: Let P be a geometric lattice of rank n. Then the Lefschetz raising operator OL in P has the property that OL,,+, is injective (i.e., of full rank) for all 0 < i < z . In addition, we conjecture that every geometric lattice is semi-Peck. Concerning the conjecture, there exist up to now only partial results, which we present now. Lemma 6.3.1. Let P be a geometric lattice of rank n and let p, q E P.
(a) Ifr(p)=1and p¢gthen q 2. Proof. (a) Clearly, q < p V q. Moreover, r(p V q) < r(p) + r(q) - r(p A q)
r(q)+l; that is, p< pvq. (b) q < v, p < v and (a) imply q < p v q < v; that is, p V q = v. (c) Since q # 1 there must be some atom p with p q. By (a), q < p V q and thus p V q # 1. Again there must be some atom p' with p' p v q. We have
q < p'vq and obviously p' V q ¢ pvq. Theorem 6.3.2 (Kung [327]). Let P be a geometric lattice of rank n > 1. There exists an order-raising operator 0 on P such that V 1, j is injective for all 1 < j
n -1 arbitrarily. By Lemma 6.3.1(c), V is well defined. In order to prove the injectivity we introduce some other linear operators
(see also Section 6.4). Let Tl, j : Nl - Nj be given by gENj:gnp=0
Moreover, let rpi := EpEN, p; that is, rpl is a fixed element of N1, and let 4)1 N1 -+ N1 be given by
01(P) =(p1 - P Finally, let a := *. Claim. Oj,Iklij j = (D 1 for I < j < n Proof of Claim. We proceed by induction on j. The case j = 1 is trivial. Since Aj+l, I `Y1, j+l = Oj, I (Aj+l, jYl, j+l ), we need only verify the equality
n-2
Aj+l,j`I`1,j+l = 411j,
for the induction step. Let p E N1, q E Nj be arbitrary. Then
(01Yl,j+1(P), q) = (`1`l,j+l (P), 0(9 )) vENj+,:v^p=0 w>q
v d (q) - I w
1
v>q:vnp=O
d -(q)
-I
0
ifpAq=p,
1
if p A q = 0
(6.23)
6.3 Modular, geometric, and distributive lattices
255
since for p A q = p, the relation v > q implies v A p = p and for p A q = 0 by Lemma 6.3.1(b) all but one of the successors of q are not related with p. The 0 same RHS appears for (tIIj(p), W), thus (6.23) is true.
It is straightforward to verify that 1 t is injective for n >_1 (note that then jV,,j and in particular Vi,j are injective. Wt > 1). Consequently, also V, = %P*
Corollary 6.3.2. In every geometric lattice of rank n there exist Wl pairwise disjoint saturated chains from level Ni to level Na_t.
Proof. Consider the operator 0 of the preceding theorem restricted to the 11, ... , n - I)-rank selected subposet. It satisfies the condition of Theorem 6.1.7, which yields the assertion. In order to generalize this result we need some facts about the Mobius function PxP-Z of a poset P introduced by Rota in [401]. The Mobius function can be defined inductively in the following way:
µ(p, p) := 1 for all p E P, µ(p, q)
-
1] µ(p, v) for all p < q, pn-k,
(p, q) = : (P (P, v), pp:vvx=1
Ifv > p, vvx = 1 andr(w) < k,thenr(v) > r(vvw)+r(vAw)-r(w) > n-k. Hence we may continue, using Theorem 6.3.4 applied to the lattice [p, 1 ], yielding
v>p:vvx=1
10
if x > p (i.e., w
1 µ(p, 1)
if x = p (i.e., w < p).
p), (6.29)
From (6.28) and (6.29) we may easily derive (c) and (with w = 0) also (b).
In addition to the Lefschetz operators we define the (linear) rank i lowering
(resp. raising) operator 0,i (resp. V,i): P -+ P as follows:
Eq
i,i(P)
(resp. V_,t(P)
Eq?P:r(q)=i q-)-
g5p:r(q)=i
ore generally, if F, Gare subsets of P then we define the operator L F, G More
F
G and its adjoint VG,F (for p E F (resp. p E G)) by
T, W (resP.GF():=
4->G(P) :_
gEG:qp /
So L,i abbreviates OP,N1. Moreover, let 0i--,j := 2iN;-_,Nj and Vi, j V Ni, N; . Note that the matrix of Di, j with respect to the bases f p : p E Ni } and { p : p E Nj ) is exactly the incidence matrix of rank j versus rank i elements.
Lemma 6.3.4.
Let P be a geometric lattice of rank n, 0 < k < n, and let
X : P x P -+ R be defined as in (627). Then for 0 < i < k,
0,; (P) = 0,; (E X (P, Y)Y) Y>P Proof. Take any w E Ni. Then by Lemma 6.3.3 (noting i = r(w) < k)
EX(P,Y)(0 Y> P
E
y> pvw
1
10
ifwi(Y)
y>q,r(y)>n-k k
X (q, Y) > 0-> i (Y) y>q,r(y)>n-k
If we apply, finally, the linear operator F : P
i=0
R defined by
F(p) := f(p) for all p E P to the LHS and RHS (noting that F (rk0
>pq,r(y)>n-k
(b) We may argue as in (a), but we may suppose that f (0) = 0 and must admit that g(1) # 0. Under these conditions, (6.31) reads
g(q) = x (q, 1)g(l)
for all q E P.
For q = 0, we obtain in particular
0= f(0) = g(0) = X(O, 1)g(1) = g(l)
> k(O, q)
p'(0' 1) r(q)?n-k
Algebraic methods in Sperner theory
260
If g(1) = 0 then, by the same reasons as in (a), f - 0. So it remains to show that
E µ(0,q)00.
r(q)>n-k
Let P+ be the lattice that can be obtained from P by identifying all elements of rank at least n - k (the truncation to rank n - k). Let µ+ be the associated Mobius function. It is easy to verify that P+ is a geometric lattice again; hence (using twice the definition of the Mobius function) µ(0, q) r(q)>n-k
57
µ+(0, q) = µ+(0, 1),
µ(0,R)
r(q) s + 1 because otherwise q* E G and thus (OF-p(ip), q*) = (AF->G('P), q* ) = 0 by (6.33). Though it may be that r(q*) > s + 1, let us first prove the statement in the theorem for this element. Take any v E P with v < q*. We will apply Mobius inversion to the interval [v, q*]. Let f, g : [v, q*] -+ R be defined by
f(x) :=
E
Ap,
g(x)
pEF:pAq*=x
E .f (Y), X E [v, q*] x 1, let P be a distributive lattice, and let fi , gi, i E [m ], be functions from P into R+ such that m
m
fj l (Pi) < rj gi i=1
i=1
M
m
t=1
i=1
VA SC(Im]) jES
for all pi E P, i = 1,...,m.
Then
flf i (Ai) < fl gi where VSC(Im)/ \jES Aj Y:p;EA; f (Pi) and so forth.
V A Aj
f o r all Al C P, i = 1 , ... , m ,
SC(17) jES
{VSC(Im)/ \jES Pj
:
Pj E Aj}, and fi(Ai)
I omit the proof of this result which is a culmination after preparatory work in particular by Daykin [ 124]. In fact, Rinott and Saks [400] proved more than Theorem 6.3.11 presenting an integral version of their result. Older versions including
continuous versions of the inequalities are those of Harris [262], Holley [271], Preston [386], Ruzsa [403], and Batty and Bollmann [39]. A further important corollary of the Four-Function Theorem, due to Fortuin, Kasteleyn, and Ginibre [185], is, historically, a predecessor of it. Theorem 6.3.12 (FKG-inequality). Let f and g be both increasing (or both de-
creasing) functions on a distributive lattice P, and let w : P -* R+ satisfy the condition
w(p)w(q) <w(pVq)w(pAq)forallp,q E P.
(6.47)
Algebraic methods in Sperner theory
268
Then
((P)w(P)) (fPPw(P))(w(P)).
(f(P)w(P))
E
pEP
pEP
pEP
PEP
Proof. We consider only the case of increasing functions. First we take characteristic functions f = q,p and g = cOG, where F and G are filters of P (WOF and VG
are increasing). Taking a := := y := 8 := w, A := F, B := G in Theorem 6.3.10 and noting that F V G is the set intersection of F and G, we obtain
(cOF(P)wP)) E
(cOGPw(P)) pEP
PEP
= w(F)w(G) < w(F v G)w(F A G)
(
< w(F v G)w(P)
=
1 cPF(P)cPG(P)w(P)
)
pEP
(
w(P)
P EP
)
and the FKG-inequality is proved in that special case. Now let f and g be arbitrary increasing functions. By Lemma 4.4.3 we can write f and g in the form k
h
g= v0+EVjcPG;,
.f =A0+XicPF;,
j=1
i=1
where F and G j are suitable filters and Ai 0, vi > 0, 1, ... , k. After a simple computation, we derive pEP (E
f(P)w(P))
=
(PEP E
xi
i1 j1
vi
g(P)w(P)) /
((E PEP
cPF
-
i = 1, ... , h, j =
E f(P)g(P)w(P) w(P) ) (PEP (PEP )
(P)w(P))
(cOGJPwP) PEP
(cOPcOGJPWP)(WP)) PEP
PEP
which is nonpositive by the special case we proved earlier.
Corollary 6.3.6 (Chebyshev's inequality). Leta1
ai bi . i=1
i=1
6.3 Modular, geometric, and distributive lattices
269
< n), define w, f, g on P by w :- 1, f (i) Proof. Let P be the chain (1 < a;, g(i) := bi, and apply the FKG-inequality. As mentioned earlier, there are many applications of such inequalities - even in domains where one would not expect it. Concerning a result of Shepp [430] on linear extensions of posets, we refer to Anderson [32, p. 100]. Recently, Ahlswede and Khachatrian [14, 17] detected a remarkable connection to number theoretic inequalities. Here we present an application to Bernstein polynomials. Let C[0, 11
denote the set of all continuous functions over [0, 1]. If f E C[0, 1], then the corresponding Bernstein polynomial B n fis defined by
(Bn f)(x) :_ E f (n) k=0
kx)nk.
(')x'(l
\
Let f, g E C[0, 1] be increasing.
Theorem 6.3.13 (Seymour and Welsh [427]). Then for all x E [0, 1]
(Bn.fg)(x) > ((Bnf)(x))((Bng)(x)) Proof. Let X E [0, 1] be arbitrary but fixed. We consider three functions w 2[n] -a 1[8+ and f, g : 2111 -). R, which are defined for all A c [n] by
w(A) := xJAI(1 -x)n-IAI,
(I n I)
.f (A)
.f
g(A) :=
g(J
,
nI).
If A C B c [n], then j (A) = f (I n l) < f(IBI) = f(B) since f is increasing. Thus f (and analogously g) is an increasing function on the Boolean lattice, that is, on 2["] ordered by inclusion. Moreover, w(A)w(B) = xIAI+IBI(l _ x)2n-IAI-IBI = xIAUBI+I AnBI (l -
x)n-IAUBI+n-IAnBI
= w(A U B)w(A n B). The FKG-inequality yields
(E f(A)g(A)w(A) I (Ag[n] E w(A) AC[n]
/
(
L, J(A)w(A) I AC[n]
/ \AC[n]
/ g(A)w(A)
Algebraic methods in Sperner theory
270
that is, `
(;)f (nk-)g( n
lIXk(1 -x)"-k/
(;)Xk(l -x)"-k)
\kE
kn (;)f(-Xk(l ) -x)n-k/ \k=o
\k/ g \k/
x)"-k I
xk(1
and finally,
(Bnfg)(x)
((Bnf)(x))((Bng)(x))
We say that a weighted poset (P, w) is an FKG-poset if for all increasing functions f and g the inequality (6.48) is true. The FKG-inequality says that (P, w) is an FKG-poset if P is a distributive lattice and w satisfies (6.47), for example, if w - 1. It is interesting that we have again a product theorem: Theorem 6.3.14 (Jones [278]). If (Pi, wi), i = 1, 2, are FKG-posets, then (PI, wl) x (P2, w2) is an FKG poset, too.
Proof. Let P := Pl x P2 and w := wl X W2. Let f and g be increasing on P. Then it is easy to see that the functions fl, gl : Pl -+ R defined for all pi E P1 by
fi(Pl)
f(Pl,P2)w2(P2), P2EP2
gl(pl)
g(Pl,P2)w2(P2) P2EP2
are increasing. Moreover, for fixed pi E Pl the functions f (pl, P2) and g(pl, P2) are increasing on P2; hence (using the fact that (P2, w2) is an FKG-poset and that w I, being a weight function, is nonnegative)
fi(Pi)gi(Pl) : (P2EP2 T f(Pl, P2)g(Pl, P2)w2(P2))
(
W2(P2)P2EP2
(6. 49)
Using the fact that (Pl, wl) is an FKG-poset, we obtain
(
f(Pi, P2)w(Pl, P2) I ((PI,P2)EP g(Pl, P2)w (Pl, P2)
/
(Pi,P2)EP
=
(
1 PiEPL
wl(Pl)gt(P1)J
wl(Pi)fi(Pt)l
/
PiEP1
1
/
6.3 Modular, geometric, and distributive lattices
1: p'EP
271
wl(Pl).fl(Pl)gl(P1)) WI(PI) I
1:
1
WI(PI) T. .f(Pl, P2)g(Pl, P2)w2(P2) I w2(P2)wl(Pl)
C P1EP1
/
P2EP2
= ( E f(P1, P2)g(Pl, P2)W(Pl, P2))
E w(Pl, P2)(P1,P2)EP
((PI,P2)EP
Let us conclude this section with a study of the variance problem for our lattices
(with weight w - 1). Because modular geometric lattices are normal, they are also rank compressed by Corollary 4.5.7. Modular lattices are not necessarily rank compressed, however. To see this, consider the subgroup lattice P of G(1,2)(p) given in Figure 6.9 and take the filter F := {U1, ... , Up, VP, G(l,2)(p)}. We have
µr(F)=p+2 5. Then by Theorem 4.4.1 the rank function is not an optimal representation if p > 5 (another counterexample was given by Stahl and Winkler (see [161, p. 84])). Moreover, geometric lattices are not necessarily rank compressed: Let us look at the Dilworth-Greene lattice DG, and take F := B (see Figure 6.7). Then n
l-tr(F) = 2 + 1
6;
that is, DGn is not rank compressed if n > 6. However, we did not expect that distributive lattices are rank compressed. More generally, we have (see [148]): Theorem 6.3.15. If (P, w) is a positively weighted, ranked FKG-poset then (P, w) is rank compressed.
Proof. Let r be the rank function of P. We verify condition (ii) of Theorem 4.4.1
in order to derive that r is an optimal representation. Let g : P = Pr -+ R be increasing and take f : P -+ IR with f(p) := r(p) - µr, p E P. Clearly, f is increasing, too. Because (P, w) is an FKG-poset, we have
1: w(p)g(p)(r(p) -Ar) PEP 1
w(I,)
7 w(p)(r(P) -
peP
since Epee w(p)(r(p) - /1r) = 0.
(PEP
w(p)g(p)0
Algebraic methods in Sperner theory
272
In Section 7.2 we will sketch a proof that the partition lattice nn is not rank compressed if n is sufficiently large. Following my paper [155], first we ask for filters F in n, that satisfy the necessary inequality µ,.(F) > µ,(11n). Because of the bijectivity between antichains A and filters F (generated by A) we study, more precisely, antichains. Let A be a positive integer. A A-coloring of [n] is as usual a function c : [n]
[A]. Given an antichain A in II, we define a proper X-coloring of A to be a A-coloring of [n] such that for each element of A (which is a partition of [n]) there exists at least one block that is not monochromatic. Let p(A, A) be the number of proper,-colorings of A. If, for example, A consists only of one partition r (with b(7r) := n - r(7r) blocks), then obviously p({ r}, A) = An - Ab(n)
Moreover, it is not difficult to see that for A = 17r, a), the equation p((n, a), A) = An - Ab(rr) - Ab(a) + Ab(nva) holds. To obtain a more general formula, we define for our antichain A the subposet LA of nn by
LA := (sup A: A' C A) (here we put sup 0 := 0, where 0 = 1121 . . . In denotes the minimal element in 11n). In the following we must distinguish between the expected value µ(.) and the Mobius function A(_). Theorem 6.3.16.
Given an antichain A in IIn, we have (0,
p(A, A) _ aELA
Proof. For any coloring c of [n], let A (c) be the set of elements of A for which all blocks are monochromatic. Further, let (p(c) := sup A (c). It is easy to see that all blocks of a := sup A' are monochromatic if all blocks of each 7r c A' are monochromatic - that is, if A' C A(c). Thus we have, for any fixed coloring c, (p (c) > LA a
if the blocks of a are monochromatic.
For a E LA, let f (a) be the number of A-colorings c of [n] with (p(c) = a. Obviously f(0) = p(A, A). Defining g(a) := L.fl>LAa f(,B), we see by the preceding remarks that g(a) counts the number of ,l-colorings for which the blocks of a are monochromatic. Hence g(a) = ,kb(,). By Mobius inversion (Theorem 6.3.3),
p(A, A) = f(0) = E µ(0, a)'X'00. aELA
0 Hence p(A, X) is a polynomial in A that is called the chromatic polynomial of A. It is a natural generalization of the chromadc polynomial p(G,,l) of a graph G (cf. [354]). If A is a set of rank-1 elements of IIn, then A can be interpreted in
6.3 Modular, geometric, and distributive lattices
273
the following way as a graph G = (V, E) on V = [n]: Each element of A is a partition of [n] with n - 2 one-element blocks and 1 two-element block, and we put ij into E if there is one element of A having ij as the two-element block. Then p(A, A) = p(G, A). Theorem 6.3.17. Let A be an antichain in fl, and F the filter generated by A. Then
(i - j)
ILr(F) ? Lr(nn) iff - E
(P(Ai)
- P(A j)) J
i,j=1
0.
Proof. We begin with the determination of p. (fu). We have I11n I = Bn (the Bell number), and, using the recurrence of the Stirling numbers (1.6), n-1
n
1: r(n) _ T i Sn,n-i = J](n i=0
7rErln
- k)Sn,k
k=1
n
E nS5,k - Sn+l,k + Sn,k_1 = nBn - Bn+1 + Bn. k=1
Thus
Bn+1
l2r(nn) = n + 1 -
(6.50)
Bn
For F 0 nn - that is, A :A {0} - we determine now E7rEF r(n). For 7r E 17n, let An := {a E A : or < 7r) and *(7r) (7r) := sup A,. Obviously, *(Jr) E LA and r > * (ir); that is, 7r lies in the filter generated by tr(ir). For a E LA, let
.fl (a) := I{n E nn : f(n) = a}I ,
r(rr).
f2(a) := 7r Erln:*(7r)=a
Obviously,
r(7r) _
.fi(a),
IF1 = aELA:a#0
f2 (a) aELA:a00
7rEF
and with gi (a) := E.8_>LAa fi (p), i = 1, 2, we have
IFI = gl (0) - fl (0),
Y' r(n)
= g2(0) - .f2(0).
(6.51)
7rEF
For fixed a E LA, the set F(a) := {8 E Fln : 6 > a) forms an induced subposet of iln which is isomorphic to ilb(a). We have F(a) = {,B E nn a} because # > a implies tr($) > a and 8 > *(f). We have gl (a) = Bb(a)
(6.52)
Algebraic methods in Sperner theory
274 because
1{rEnn:*(n)=P}I
gl(a) _ ?:LA a
= I{7r En, : r(7r)>a}1=lnb(a)I=Bb(a), and
(6.53)
g2(a) = nBb(a) - Bb(a)+l + Bb(a) because
>LAa 7rEn.:* (7r)=#
n-1
r(n)
r(n) _
r(n) _
g2(a) _
7rElln:7r>a
7rEI1n:1,(7r)>a
b(a)
iSb(a),n-i = 57(n - k)Sb(a),k = nBb(a) - Bb(a)+1 + Bb(a) i=r(a)
k=1
where the last equality follows as in the beginning of the proof. From (6.51), (6.52), and (6.53), we obtain via Mobius inversion
IFI = Bn - E µ(0, a)Bb(a),
(6.54)
aELA
r(7r) = nBn - Bn+l + Bn 7rEF
(6.55)
- L..: µ(0, a)(nBb(a) - Bb(a)+l + Bb(a)). aELA
From (6.50), (6.54), and (6.55) we obtain after a straightforward computation that
Ar(F) > /Lr(nn) if
(6.56)
µ(0, a)(Bb(a)+l Bn - Bb(a)Bn+l) > 0. aE LA
By Dobinski's formula (1.7) a product Ba Bb can be written in the form 1
1
1
BaBb=2(BaBb+BbBa)=2e2
is b
ib a
i,l +il i,j=O 1
Thus (6.56) is equivalent to 00
E (i - j) tl
i,j=O
1
'
t
jn E µ(0, a)ib(a) - in aELA
µ(0, a)jb(a) aELA
0,
6.3 Modular, geometric, and distributive lattices
275
and the last inequality is by Theorem 6.3.16 equivalent to 00
i) lnn (i - j) (p(A, in
l= l
i!j!
\
- p(A, j)) > 0. n
J
We call an antichain A of fl, coloring-monotone if
p(A,A) < p(A, ; + 1) (A + 1)n
a.n
for all A E N+.
Hence A is coloring-monotone if the probability that there exists a nonmonochromatic block for a random coloring in each element of A increases with the number of colors. Directly from Theorem 6.3.17 we may derive:
Corollary6.3.7. Let A be an antichain in nn and F the filter generated by A. If A is coloring-monotone then
iir(F) ?: µ,(n,,)
Proposition 6.3.2. If I A I < 3 then A is coloring-monotone. Proof. The cases I A I E 11, 2} are easy to verify, so let A = {a, 0, y). Generalizing the case IA I E 11, 2} we may calculate p(A, A) also by the principle of inclusion and exclusion:
p(A, A) _ An - a,b(a)r(a) - Ab(P)r(f) -;,b(Y)r(Y) + kb(avfl)r(a V fi) +.kb(avy)r(et V Y) + ab(Pvy)r(8 V y)
- Ab(avflvy)r(a V P V y).
We put f o,) := p(A,
It is sufficient to prove that f'(),) > 0 for ,l > 2.
We replace everywhere b(ir) by n - r(n). In the calculation of f'(,k) we omit the (positive) term with a V # V y and decrease ,l-r(a) by A-r(a\ ) + A-r(avy) (note
that r(a) + 1 < r(a V 0), r(a) + 1 < r(a v y), A > 2) and do the same for ),-r($) and X-r(y). We obtain
/A-r(avf)(r(a) f'(A)
+ r(p) - r(a V ))
+,1-r(avy)(r(a) + r(y) - r(a V y)) +.l-r(RvY)(r(p) + r(y) - r(P v y))) The RHS of this inequality is nonnegative because the partition lattice is semimodular.
Algebraic methods in Sperner theory
276
I leave it to the reader to verify for several classes of graphs (see [61] or [354]),
like complete graphs, trees, circuits, wheels, interval graphs, and so forth, the coloring-monotonicity (recall that subsets of Ni (II,) are interpreted as graphs). We studied the case I A I = 1 together with Bouroubi [82] in a direct way. Let me end with the remark that not all antichains are coloring-monotone if n is sufficiently large. Take for simplicity n even and A C N1, whose two-element blocks have the
form {i, j} with i < z, j > 2 (this can be interpreted as the complete bipartite graph on the vertex set 11 , ... , 2 } U J!! + 1, ... , n}). Then it is easy to see that p(A, 2) = 2 and p(A, 3) = 6 + 6(2 - 2); that is, p(A, 2) < p(A, 3) iff 3n-l < 2 2n
3n
- 2",
and this is false if n > 20. Here is a nice analogy to the proof of Theorem 5.4.8.
6.4. The independence number of graphs and the Erdo"s-Ko-Rado Theorem In this section we consider more general objects, namely simple graphs G = (V, E) instead of posets. Let us recall that a subset S c_ V is called independent if vw 0 E for all v, w E S. The independence number of G is defined by a(G) := max{ISI : S is independent}.
The determination of a(G) is, in general, difficult (NP-complete, cf. [215]), but here we are looking at special graphs that are, in particular, regular. We will consider an algebraic method for the the determination of upper bounds for a (G). It is based
on the work of Hoffman [119, p. 115], Delsarte [128], Lovasz [355], Schrijver [419], Wilson [470], and others. Though there are theories related to these questions - the theory of association schemes, cf. Bannai and Ito [38], and the theory of graph spectra, cf. Cvetkovic, Doob, and Sachs [119] (see also Haemers [250]) - we shall avoid building the whole machinery. Instead we'll examine some main ideas that
are sufficient for our purposes. As throughout this chapter, linear operators are preferred to matrices. _ We associate with a graph G the graph space G in the same way as it was done for posets (see Section 6.1). So G is freely generated by the vertices of G. The standard basis of Z Y4 given by {v : v E V}. Moreover, we consider G as a Euclidean space (with the standard scalar product). We will work with (linear) operators from G into G. The identity operator on G is denoted by I. Let
IPG:=Ev. vEV
We define the all-one-operator .7 by
J(v) := coG for all v E V.
6.4 The independence number
277
G is said to be an adjacency operator if for all
Finally, an operator A : G
vEV
A(v) _ E c(v, w)w, wEV:VwEE
where c(v, w) are real numbers depending on v and w. Note that (A (U), w) = 0 E. If for all vw E E there holds c(v, w)_= c(w, v), then (A(v), w) _ if vw (v, A (w) ); hence A is self-adjoint. For example, J is self-adjoint. If c(v, w) = 1 for all vw E E, we speak (as for posets) of the Lefschetz adjacency operator AL. It is well known that all eigenvalues of a self-adjoint operator A are real and that there exists a basis of G consisting of pairwise orthogonal eigenvectors of A. Recall that a self-adjoint operator M is positive-semidefinite if (M(op), cp) > 0 for all cp E G and that this is equivalent to the fact that all eigenvalues of M are nonnegative.
Lemma 6.4.1. _ Let A be _a self-adjoint adjacency operator on a graph space G
and let M := I + A - 1,7 be positive-semidefinite. Then a(G) < S. Proof. Let S be any independent set in G and let cos := EVES v. Then
iu' _ E (A(v), w) = 0
(A(ops), cos) _ (1: A(v), VES
WES
V,WES
since vw V E for all v, w E S. Consequently,
0 < (M(ws), ws) _ (cPs, ccs) + (A(ws), ws) - s (J(ws), ccs)
=A-
(J:.7(V),,os) VES
= I SI -
I SI 4G, ws) = ISI -
s
I SI2
This implies ISI < S. Remark 6.4.1. Note that in Lemma 6.4.1 a(G) = S iff (M(ops), cps) = O for some independent set S in G. Up to now we have had much freedom in the choice of A, but we must choose A and S such that the verification of the positive-semidefiniteness of M is possible. One way to do this is the construction of A as a linear combination of "elementary" adjacency operators for which the eigenvalues can be determined explicitly: Lemma 6.4.2.
Let A = >j"_ i fi j Bp where each Bj is a self-adjoint adjacency operator on the graph space G of a graph G on n vertices. Suppose that the eigenspaces of the operators Bj are independent of j, and let ?. j,,, i = 1, ... n, be
Algebraic methods in Sperner theory
278
the eigenvalues of Bj, j = 1, . , 1. Finally suppose that cpG is an eigenvector of Bj to the eigenvalue ;lj,l, j = 1, ... ,1. Then the eigenvalues µi of M := I +
A - s J- are given by
ifi=1, ifi =2,...,n.
1+:j=1PJ),J,1 1+
Proof. By our suppositions, there exists a basis {cpl = cpG, cot, ... , cp } of pairwise orthogonal eigenvectors of B j to the respective eigenvalues X j, 1, . . . , A j,,, j =
1,...,1.We have J(cpG)=EVEVJ(v)=EVEVEWEVncpGand J(cpi)= 0, i = 2, .. ., n, since for each v E V it holds ((pi, .7(v)) = (0, cpG) = 0. Consequently,
M(cpG) = VG +PiBj(coG) - J(cpG) _ (1 +pjXj,1 - n VG,
\
j=1
l
1
M(cpi) = cpi +
E
J
fijxj,i 0, i = 2, ..., n.
BjBj(cpi) - 1 J(cpi) = 1 + 8
j=1
j=1
)
With the assumptions of Lemma 6.4.2 we obtain the smallest possible upper bound on a(G) in determining
z:=max Epjxj,t j=1
subject to
i =2,...,n.
3jAj.i > -1, j=1
Then we have µi > 0 for i = 2, ... , n and we may choose n
+z (this is the smallest S with the property µt > 0). We summarize the result: Theorem 6.4.1. Let B1,..., BI be self-adjoint adjacency operators on G whose eigenspaces are independent o f the index. Let A,,1 (i = 1, ... , n = IV (G) I), be the eigenvalues of Bj, j = 1, . . . , 1, and let c°a be an eigenvector of Bj to the eigenvalue of Aj,1, j = 1, . . . , 1. Define z := max{Ejl - -1 1jAj,l : El -J'-1 ijj,i > -1, i = 2, ... , n). Then
a(G)
2n, we have on both sides polynomials in z of degree at most n that are equal on n + 1 points, namely z = qn, ... , q2n; hence they are identical. For fixed z and a such that a > 2n, we may conclude in the same way that the assertion holds for all y, z and then finally also for all x. Corollary 6.4.2.
We have
(a) xn = 1 + rk=1 lk)q(x - 1)(x - q) ... (x - qk-1), (b) (x - 1) (x - q) ... (x - qtt-1) = rk=o lk)q (-1)kq (2)xn-k,
() l(a+b)q = k=0 n
n
(d)
(b-a) =
ln
q
((b (a-k)(n-k) In-k)qg q-ak-(2)-(a+k)(n-k). rk=o(-1)k a+k-1 / b ((a
k
)q In-k)q
Proof. (a) Put in Theorem 6.4.2 z := 0 and y := 1. (b) Put in Theorem 6.4.2 z := 1 and y := 0 and use k := n - k. (c) Put in Theorem 6.4.2 x qa+b, y qa, z := 1 and use k := n - k. (-1)kq-ak-(2) (d) Replace in (c) the variable a by -a and note that ( k)q = * (a+k-11 l
k
)q'
As always we will use the letter P for both lattices Ln (q) and Bn, and r denotes the rank function. Elements of P are denoted by X (subspaces of an n-dimensional vector space V over G F(q) (resp. subsets of the n-element set [n])). The standard
6.4 The independence number
281
basis of the poset space P is then {X : X E P}. Recall that r(X) = dim X (resp. XI) and that a family.F C P is called k-uniform t-intersecting if r(X) = k for all X E .F and r(X A Y) > t for all X, Y E Y. Let G = G,,,k,t(P) be the graph on the vertex set Nk whose edge set E is given by E {XY : X, Y E Nk and r(XAY) < t}. We call G briefly the Johnson graph because it arises in the Johnson scheme (see [38]). In the case t = 1 our graph G is called the Knesergraph. Clearly a(G) equals the maximum size of a k-uniform t-intersecting family in P. Our aim
is the determination of a(G). (In the case of B we know already the solution by Theorem 2.4.1. We will derive this result again for n > (k - t + 1) (t + 1) since the method gives in addition many interesting properties of the corresponding linear operators.) We may suppose throughout that 0 < t < k < (n + t)/2 since otherwise the problem becomes trivial. We have
a(G) > (n k
t) t
(6.57)
q
since we may take a fixed X0 E P with r(Xo) = t and obtain a k-uniform tintersecting family S := {X E Nk : X D Xo} of size (k-t)q .The question is for which parameters we have equality. We will see that this is the case if n > 2k for
(t+1)(k-t+1)forB,,.
In order to define the operators Bj of Theorem 6.4.1 (not to be confused with the Boolean lattice for which we write P here) we work with the operators
Vi, j and Di, j introduced after (6.30). Recall that, for example, Di, j(X) :_ EYCX:r(Y)=j Y, where X E Ni. Moreover, in generalizing the operators from the proof of Theorem 6.3.2, we introduce tPi, j : Ni -+ Nj as follows:
'Pi, j(X)
where X E Ni,
0 < i, j < n.
YENj:XAY=O
Finally we define Bj : Nk - Nk by Bj
WJ kOk,j,
.1 =0,...,k.
Lemma 6.4.3.
(a) Let X E Nk, Y E P, r(X A Y) = m, and 0 < j < k. Then I {Z E Nj : Z
j = 9
(e) Pi->l =
- j < i - n,
lj-l)gVi-.l, 0 < 1 < j < i < n, gi(i-1)(nT
j
1
Ej_0(-1)jq(j}2
)
0 < 1 < i < n.
Proof. (a) and (b) follow from the fact that for X > Y, r(X) = i, r(Y) = 1 it holds I (c) and (d): With a method analogous to the proof of Lemma 6.4.3(a) we obtain
that, for fixed Y E N1, X E Ni with Y A X = 0, the equality I{Z E Nj : Y < Z, Z A X = 0}1 = qi(j-1) holds. This yields the assertion. (e)LetXE Ni, Y E N1, r(XAY) =m. We have
(Vp ii i_, j(X), Y) = IfZ E Nj : Z < X and Z < Y}l
={ZENjZ<XAY}l
\j/q.
=
Consequently, putting in the polynomial identity from Corollary 6.4.2(a) x := 0 and n := m we obtain min(i,l) _1 m
(-1)iq(z)pj,101-, j=0
jl (X), YI = (-1)Jq(z) (in') 1
j=0 1
10
\ )q
ifm =0, otherwise
= hpl-,l(X), Y)
6.4 The independence number
283
(f) Let X E Ni, Y E N1, r(X A Y) = m. We have
(Oj,l,Pi,j(X),Y) = I{Z E Nj : Z < YandZAX=0}I
= I{ZENj:ZkOk-s1) _'Pj,k(Ok,j*l-+k)Ok-+l
=
ql(k-j) (n -
j - 11
1\ k-j Jq
4'j-.kT1-
jOk-+1
)q(i;)qBi.
Algebraic methods in Sperner theory
284
and further l}
qjj,k'Fl-->jAk,I = Wj-->k min{ j,l }
r=0
(mt
(-1)'q(2)Qi j(0! si0k >!)
(- 1)'q(2) (k-i)
min{ j,1)
-1)'q i=O
((') k-
k(j _i) (n - k - i
\ ji )q'i+kt
i qq
m i n(j, 1 }
E
9
(-1)igk(J-i)+(2)
1=0
k+i
n-k-1 k-i i. j )q (1-i)q B
The product B1Bj can be calculated analogously.
Lemma 6.4.6. Let 0 < j < k < n and j + k < n. Then
(a) 4 j-+k is injective,
(b) Bj(Nk) = Vj-rk(Nj). Proof. (a) By Lemma 6.4.4(c) and (f) we have
j Oj_j = L(-1)'q(12')
_
ii
i=0
(±(_l)jqc)_jk(fl +i
k
-)' ijki) Wjk i
i)
Since 0 j.+j is the identity operator on Nj our operator W j ,k must be injective. (b) By definition and Lemma 6.4.4(a) and (e) we have
$J =
j ,k k,j
i
',(--1)'q(2)
=
k-i
j -I q
i=0
((_1)1q(ijki), !=0 hence
Bj(Nk) c Oj->k(Nj).
(6.58)
Moreover, since k4 j is a scalar multiple of the Lefschetz lowering operator restricted to Nk, we derive from Theorem 6.2.5 that Ok,j is surjective and Vj +k
is injective. Consequently (using (a)), dim Bj (Nk) = dim Tj,k(NJ) = Wj = dim Vj,k(Nj). This gives equality in (6.58).
285
6.4 The independence number
Let us look at the spaces E11 defined in (6.14). For our purposes, we use here other indices. Recall again that Disk and Di-+1_1 are scalar multiples of OLr.A (resp. AL11_1). Consequently, we have
0 < i < k,
Ek,i = Ol->k(ker(0,-->1-1)),
(6.59)
and by Theorem 6.2.5
0 i. Since iP E V1->k (Ni ), there is by Lemma 6.4.6(b) some ip" E Nk such that Bi (cp") = cp. We first show that, for 0 < I < i, there holds Bi (ip") = 0: In view of Lemma 6.4.5 and Case 1 we have 0 = Bl (cp) = B1 Bi (cG") = Bj BI ((p").
Again by Lemma 6.4.6(b) there is some iP"' E N1 such that Ol->k(ip...) = Using the definition of Bi, we conclude that
0= But Ok,iVl-+k is injective according to Theorem 6.2.5 (note 1 < i < k) and Wi.+k is injective by Lemma 6.4.6(a). Consequently, V'I' = 0 implying BI (cp") = 0. This
Algebraic methods in Sperner theory
286
result and Lemma 6.4.5 imply Bj (co) = Bj Bi (c'") =
=
gj(k-i)
Bi Bj (tp")
C k)q(n-k_i) k-i) n - i - j r(-1)igk(i-i>+(z)
q1(k-i)(n - i
-j) (-1)`q(i)(j-i
k-i
q
k-i
B1(")
\>
cp
q
For the application of Theorem 6.4.1, we need further that cPG = >XENk X is an eigenvector. This is in the present situation the case because coG generates Ek,o. As mentioned previously, The corresponding eigenvalue is X j,o := qjk (" k J)9 9 we will work with an operator of the form A = ,Bk-r+1 Bk-t+1 + + iBk Bk. Here we need numbers ,Bj, j = k - t + 1, . . . , k, such that k
Pjaj,i > -1,
i = 1, ... , k,
j=k-t+l and Ei=k-t+l Pj'kj,o is as large as possible. To get an idea how to choose these /3 j's, let us look back at the beginning of this section. We are looking for parameters n, k, t such that a(G) = (k-t)q So we take in Lemma 6.4.1 8 := (k=i)q Independent sets S of this size were represented after (6.57). The corresponding vector in Nk has the form cps = >XENk.X_Y X, where Y E Nt is arbitrary.
Obviously, cps = Ot_k(Y). By Remark 6.4.1, a(G) = 8 and S is maximum
if (M(cPs), cos) = 0.
Moreover, M has to be positive-semidefinite. Hence the vectors cos are optimal solutions of the problem (M(cp), cp) -> min, where cp E Nk. This implies that M(cps) = 0 (either expand cos as a linear combination of eigenvectors or use the fact that the gradient of the corresponding quadratic objective function must be the zero vector). Let X E Nk and Y E Nt be arbitrary. From the preceding remarks we obtain the condition
X) = 0; that is,
(Y, Ak_rM(X)) = 0.
6.4 The independence number
287
We have in view of Lemma 6.4.4(c)
j = qi(k-t) (n_j_t k -tt
Ok->tBj =
IPjItAk->j.
Hence k
Ok->tM = Ok-*t +
pjgi(k-t) (
j=k-t+l
n-k I t )t tl'j-tLk- j - s 1Ok- rJ. )q
1\
Let m := r(X A Y). Then
(Y, Ak,t(X)) _
(1
if Y < X, i.e.,m=t,
0
otherwise,
(Y,4j.._,tOk,j(X))= I{ZENj:Z<XandZAY=O}I =
qmj(k-m) j
9
(this follows from Lemma 6.4.3(a)), and
(Y, Ok,tJ(X)) = I {Z E Nk : Z > Y}I
= (k -
t) = S. 9
Accordingly,
0 = (Y, Ak-tM(X))
_
if m=t,
11
k
P qj(k-r)
+
0
otherwise
(n - j - t
j=k_t+l
k- t
k-m qmj j
q
q
So we transformed our condition into the following system of equations: k
pjgj(k-t+m) n-
j-t
k-t
j =k-t+1
_
)q(k_rn) 3
q
0
ifm = t,
1
if 0<m 0 and j + 1
(tj
i - t+
k-tj; j
1)('t-t+
i - t+
6.4 The independence number
293
Direct computation yields
1)(n-k+j-k)
ki
Ckkl t j)(n (ti--1I)(t-iI)(k-il-I
j)(n-l+j-k)-1
Consequently,
Bi =
(-1)t-1-i(t-1
JCtj1)(k111,)Cn-k-t+j)-1
E (-1) j=p
(6.62)
Consider the function
f = f (x, y; a, b) := b
Let If := f and
with x, y E R, a, b E N, y > b.
(a) (b) -1
Xf (x, y; a, b) := f (x - 1, y; a - 1, b),
Yf(x, y; a, b) := f(x - 1, y+ 1; a, b + 1),
Zf(x,y;a,b)
f (x - 1, y + 1; a, b).
In a straightforward way one may verify that
XYf = YXf, (I - Z) f = (X + Y) f. Accordingly,
(I - Z)t-lf = (X+Y)t-If; that is,
t - 1)Z j f j=0 t-1
j j(t-1) 1\
t - 1)Xt-1-JYjj,
j=0
j
j=0
=l(t-1) f(x-t+l,y+j; a-t+1+j,b+j). j=0
j
If we put here x := k - 1, y := n - k - t, a := i - 1, b := i - t, then the LHS becomes the same sum as in (6.62) divided by i - t, and the RHS becomes the sum in the assertion divided by i - t.
Algebraic methods in Sperner theory
294
Lemma 6.4.10.
Let l < t < i < k < z and n > (t + 1) (k - t + 1). Then
0i>-1. Proof. From Lemma 6.4.9 we know that 0t+1, t + 3 , . . . are nonnegative and that Ot+2 increases (let+21 decreases) if n increases. Lemma 6.4.8 says that 0t+2 > -1
(if t < k - 2 and n > (t + 1)(k - t + 1)). So it is sufficient to show that IOiI> IOi+1I
fori>t+2.
We work with the formula from Lemma 6.4.9. If t = 1, then ik-1`in-k-1` k-i lei+l I _ i-1 i 2k. Thus let t > 2. We show that the ratio of the summands with index j (j < min{k - (i + 1), t - 1}) in the formulas for Oi+l and Oi is smaller than 1. We have It//i
1) (i + 1 - t)
\(t
j 1)(i+1-t+ j) (i - t +/ j) In i k-t )
It-1)(i + 1 - t+ f) (i+1-t+j)(i - t)(t j1(i kt+j)
i -t+j
i(k - i - j)
i+1-t + j (i - t)(n - k - i)
\
i(k-i)
(i-t)(n-k-i)
We have further
n - k - i > ((t+1)(k-t+1)-k-i)-(t-1)(i-(t+1))=t(k-i), and, because i - t > 2, t > 2,
(i-t)t>(i-t)+t=i. Hence we may continue the estimation of our ratio:
i(k - i)
i(k - i)
(i-t)(n-k-i) - (i-t)t(k-i) 15). Schrijver [419] had come close to the solution before (using Delsarte's and Lovasz's ideas).
6.5 Further algebraic methods
295
Of course, the presented method can also be applied to other structures. See, for example, the papers of Stanton [445], Moon [373], and Huang [274].
Finally, note that one may derive with this method that, for n > 2k (resp.
n > (t + 1) (k - t + 1)) in the case of L (q) (resp. B,), every maximum kuniform t-intersecting family S has the structure S = {X E Nk : X > Xo} for some Xo E N. This can be shown using Remark 6.4.1. Strict inequalities in our for every maximum family and then estimates imply that EXES X E an induction argument yields this structure. We avoid detailed presentation and refer to Wilson [470].
6.5. Further algebraic methods to prove intersection theorems As in the previous section, we will regard in the following the Boolean lattice B and the linear lattice L (q) simultaneously and we will use the letter P for both
B and L,, (q). We have formally limq.1 L (q) = B. Recall that for a family F in P, .F denotes the subspace of P generated by {X : X E .F}. Assume that we have a second family 9 in P. In this section we will often work with the operator
O,F,g :.F
9 and its adjoint Vg. f : G -> F, which were defined in (6.30)
by
Y
YEc:Y<X
(resp. Og (X) :_
\
Y YE.F:X family
k1 >...>kr> ILI-r+1.Then I.FI
r
n
(lLI1_j)q The same result is true for B if we let q - 1. Proof. We take
U N1 NIL I+l - j , A and (D as in the proof of Theorem 6.5.1.
If i > ILI + 1 - r then Dj +r(Ni) c Vg+.(C). Thus let i < ILI - r; that is, i < kr. Recall that Tk, :_ {X E Y : r(X) = kj}. Claim. There are numbers Pi, l = 1, ... , r, such that for 1. E1=1 81( Before proving this claim, let us first finish the proof of the corollary. We have
for X E Ni (X) j=1
r
kj - i
- j=1 1=1o1(ILI r
_
1-1-i)9
X (56
r
>2 > P1V jLI+1-1->.F'k1 ViILI+1-1(X) j=1 1=1
j=1
= 09-sF
!=l
(ElIL+1_1) C 1=1
which implies as previously A (.j5) c (note that the summands with 1 < I L I + 1 - kj are zero (resp. the zero vector)). Proof of Claim. We must prove only that the columns of the matrix of our system of equations are linearly independent. Assume the contrary. Then there exists a nontrivial solution of the system Y/.fa-!(qhl) _ O, 1=1
j = 1, ... , r,
(6.66)
6.5 Further algebraic methods
299
where
fm (x) :=
(x - 1) ... (x - qm-1) - 1) ... (q m - q m-l )'
(q "'
a :_ ILI + 1 - i ,
hi
:= k-- i
(x -m + 1) (in the Boolean case we have to work with the functions m, x (x -1) qa-r-l ELI - r + 1, we have qhi > and to insert hj instead of qhi). Because kj a I )-we obtain that then there for all j. Dividing (6.66) by (qhi -1) ... (qhj also exists a nontrivial solution of the system
-q
SO + S1(ghi - qa-r) + ... + Sr-1(ghi - qa-r) ... (qhi - qa-2) = o,
But this is impossible since the nonzero polynomial 80 + 31 (x - qa-r) + Sr_1(x - qa-r) . . . (x - qa-2) can have at most r - 1 roots.
+
Under some injectivity suppositions we may estimate shadows in B (Frankl and Fiiredi [194]) and L (q) (Frankl and Graham [198]): Theorem 6.5.2. Let 0 < s < I < k < n, T- a k-uniform family in L (q) and let A.F,,s be injective. Then k+s I )q (k+s)
ls
q
The same result is true for B if we let q -* 1.
Proof. Let P = L (q) (resp.
For s = 1, we may use the standard observation
that
ICI = dimOF, (9) < dim A " (Y) = Ii -->s(f)I, and for k = 1, the result is trivial. We apply induction on k and suppose that s < 1. Let us fix some element Y in N1. The filter generated by Y is again a linear lattice (resp. Boolean lattice) with the parameter n - 1. We denote this filter by Py. Moreover, let.T'y := F fl Py. The notations AY, Dy, and soon mean that we consider the shadow (resp. shadow operator) only in Py (resp. Py). Observe that .Fy is a (k - 1)-uniform family in Py and that N5 1 fl Py is the sth level in Py which we denote by Ns y. Claim. L ry..+s is injective. _
Proof of Claim. We know that A,,,,s is injective, which implies that also O7Y,s is injective; that is, surjective, and this is the case if Ds-. Y (X)
is surjective. It suffices to show that OS fY is
E VS (Ns)
for all X E NS.
(6.67)
Algebraic methods in Sperner theory
300
If X V Py then Vs->.ry(X)
= V .jy(XV Y)
since for Z E Fy we have Z > X iff Z > X V Y. If X E Py then as in Lemma 6.4.4(b)
Os->.-y(X) =
Y Y Os-,.FyVS-I"(X).
1
k-l-(s-1)
(
l
)q
Thus in both cases (6.67) is proved.
We may apply the induction hypothesis to the (k - 1) -uniform family Fy in Py to infer
k-l+s (6.68)
l+s) q
10Y+1-1(.FY)I > IFYI
k-l )q If we count the number of pairs (Y, X) with Nl D Y < X E .P in two different ways we obtain that IFYI =
(k)
YENI
(6.69) q
and counting the number of pairs (Y, X) with Nl D Y < X E &+1(.T') gives
E IAY,i-1(FY)I = (') YEN1
(6.70) q
Summing up (6.68) over Y E N1 yields, with (6.69) and (6.70),
k l+s
( 9
k-1
q
which is equivalent to the statement in the theorem. Note that the bound in Theorem 6.5.2 is the best possible. Taken := k + s, F := Nk and note that Ok-+s is injective since and B are unitary Peck (see Example 6.2.9). Part (b) of the next corollary is a result of Katona [290] in the Boolean case, which was proved by him without the algebraic approach. Frankl and Graham [198] also considered L (q).
Corollary 6.5.3. 1 < k. Then
Let .P be a k-uniform t-intersecting family in L (q), k - t
I
The same result is true for B if we let q -+ 1.
6.5 Further algebraic methods
301
Proof. (a) Obviously .T' is L -intersecting where L = [t, t + 1, ... , k -1 } ; that is, I L I= k - t. From the proof of Corollary 6.5.1 it follows that Ok-t..+ is surjective, because the corresponding operator A F is injective - that is, bijective.
Consequently, O.'+k-t is injective. Because (1-k+t)q &F-.k-t = Zl-+k-t therefore O.F->l is also injective.
(b) From (a) we know that 0.F,k-t is injective. The result follows from Theorem 6.5.2 with s := k - t. The following classical theorem is due to Katona [290] in the Boolean case. I treated the linear lattice for t = 1 in [161], and Lefmann [337] generalized this to arbitrary t. Theorem 6.5.3. Let .T' be a t-intersecting family in L" (q), t > 1. Then if n + t is even,
Ek>M? (k)q
IFI < n+t-I
q
+
k> !±j±
Dq
if n + t is odd,
and the bound is the best possible. The same result is true for B" if we let q -+ 1. Proof. Under all maximum t-intersecting families in L" (q) (resp. B"), we choose a family for which k* := min{k : Fk 0} is maximal. If k* > (n +t - 1)/2, we
are done. If k* = (n + t - 1)/2, we only have to prove Claim 1..T'"+t-I)/21 // n If k* < (n + t - 1)/2 we will construct below a new t-intersecting family F' with IF'I > IFI and min(k : Fk' 0- 0) > V. This is then a contradiction to the choice of F. 1
For k* < (n + t - 1)/2, let
c:_{XEN"+t-k*-I:r(XnY)=t-1 forsome YE.Tk*}. Because F is t-intersecting,
Fnc=0. Claim 2. We have 2k*-t
lk*-t+l)q
ICI
- (2k*-t) IFk`I k*
Proof of Claim 2. Let 7l :=
q
(Fk*) We have
{XEN"+t-k*-t : there is some Z E H with X A Z = 0)
(6.71)
302
Algebraic methods in Sperner theory
because for X, Z from the RHS the existence of some Y E Fk* with Z follows, and we have r(X v Z) = n; that is, also r(X v Y) = n, hence
Y
r(XAY) = r(X)+r(Y)-r(XVY)
_ (n+t-k*-1)+k*-n=t-1. Recall the definition of W;+j before Lemma 6.4.3. Obviously
tk*-t+1-sn-k*+t-1 (x) C Q. Because Wk*-t+1-sn-k*+t-I is injective by Lemma 6.4.6(a), it follows that 191 >- In l.
Moreover, by Corollary 6.5.3(b), /2k*
t
I HI '- t2k*-t k*
IFk* I. q
0 Proof of Claim 1. For Ln (q), the assertion follows directly from Corollary 6.4.3. For Bn, we work with k* :_ (n + t - 1)/2 and 9 from above. Because of (6.71) IJ7 k* I + IQI < (k*
and in view of Claim 2 we infer
l"" (n-1
n+t-I
tt
l
which is equivalent to the assertion.
For k* < (n + t - 1)/2, let '' := (.F-.Fk*) UC9. Because of (6.71) and Claim 2 (note k* - t + 1 + k* > 2k* - t) we have IF'I > I.FI (by the way, this can be proved also in an elementary way, see [337]). Thus it remains to prove the third claim. Claim 3..F' is t-intersecting. Proof of Claim 3. Let X, Y E F'. If X, Y E .F, then r(X A Y) > t because .F
is t-intersecting. If X E F, Y E 9, then r(X A Y) = r(X) + r(Y) - r(X v Y) >
k*+1+n+t-k*-1-n = t. IfX,Y E c, we have also r(XA Y) _ r(X)+r(Y)-r(XVY) = 2(n+t-k*-1)-n > n+2t-(n+t-1)-2 = t-1.
6.5 Further algebraic methods
303
To see that the bound is the best possible, fix some Y E Ni_1 and take
{X : r(X) > "zt }
if n + t is even,
I] I {X:r(X)> "+t+ U { X E N - , :X0). Theorem 7.1.2. Let 1;1, ... , + fin. Then '1 +
n
be independent random variables and let
(p
1
(t)... (P,(t).
Corollary 7.1.1. If 1, ... , In are independent random variables with the common distribution F, mean µ and variance a2 then n = 1 + + l;'n is asymptotically normal with mean nµ and variance nat.
As an example we consider the poset P = S(k, ... , k) which is a product of n (k + 1)-element chains (the succeeding result may be generalized in a straightforward way to a product of n copies of any finite ranked poset). Each such (k + 1)-
xj; hence element chain C has the rank-generating function F(C; x) _ F(P; x) = Ejko Wjxj = (F(C; x))n. We introduce the discrete independent random variables j, i = 1, . . . , n, which take on the values 0, . . . , k each with probability
k+1
.
Let l; be one of i;1, ..., 4'n . Note that
tP (t) =
Moreover, n := 1 +
j=0 k + 1
e`jt
e"). = k + 1 F(C;
+ n takes on the values j = 0, ... , nk each with
probability Wj (P)/(k + 1)n. We have indeed
tP (t) =
(F(C; eit)/l
n
=
1
(k+ 1)nn
F(P; eit).
By Corollary 7.1.1, n is asymptotically normal with mean ttn := n i and variance
an := n (k(k + 2)/12). In particular it follows that yyj ^ (41 (b)
- 0 (a))(k+ 1)",
(7.1)
Qn a+µn :Sj 0 and put y := 4, S 2. From (7.3) we obtain for sufficiently large n:
E LAnJ E. We study only x > E. Using (7.3) and the just proved monotonicity of the an (k) for k > An + (x - 8)an we derive (Lµn + xanj - Lµn + (X - 8)anj + 1)an(Lµn + xanj)
an(k)=fi(x)-fi(x-S)+o(1)
E
< Li
(7.4)
n+(x-S)onJ 0 1
_
n!
ez
1
27ri
IzI=r
Zn+l dz.
Substituting z = rein, we obtain rn
'r
1
n I - 2n f,r a
re'-nip d(p.
We divide the integral on the RHS into two parts Il + 12, where
Il =
f
otn)
and
ere"r-nicpdco.
12 = S(n) 0 such that
max Ih(z)I = Izl=r
O(f1-6(r)),
r -+ oc,
then f (z) + h (z) is admissible. One can show that, in particular, f (z) + p(z) is admissible.
Construction 3: If the leading coefficient of p(z) is positive, then p(z) f(z) and p(f (z)) are admissible.
7.1 Central and local limit theorems
313
Let f (z) be an admissible holomorphicfunction with Taylor series f (z) = En° O anzn. Let a (r) := r 'r)) = r dr log f (r) and b(r) := ra'(r). Let rn be the largest root of a(r) = n (which can be shown to Theorem 7.1.8 (Hayman [265]).
exist). Then
f(rn) asn -+ o0.
1
an
27rb(rn) rn
We will not develop here the theory of generating functions (cf. Aigner [21], Stanley [441], and Wilf [469]). As an example, we consider the Stirling numbers of the second kind.
Lemma 7.1.1.
We have Sn kYkzn nI n,k=O
=
ey(ez-1)
Proof. Obviously, eyW-1)
(ez - 1)k
k
00
k
k=O
1
00
=kE
Z2
Z2
\z21...1... (z2i+...1
Y
ki
00
= L k,n=O
1
1
1l+...+Ik=n
kZn
il!...ik!
il,...,ik>l
Now the assertion follows from the fact that
k!Sn,k = I (f : [n] - [k] : f surjective}I
E
I{f: [n] - [k]: If-1(1)1 =i1,..., If-1(k)I =ik}I
i l +...+ik =n
iI,...,ik>1
E
i I +...+ik=n
n!
i11...ik!
il,...,ik>1
If we set y := 1, we obtain 000 B
L: n. n=0
zn = ee -1.
Limit theorems
314
Obviously, this function is admissible. Thus we obtain from Theorem 7.1.8 for the size of the partition lattice 11,, ee'-1
1
Bn
n!
where re' = n.
27r(r2 + r)er rn
Finally, using Stirling's formula (Theorem 7.1.7) and er = ?, we derive a formula of Moser and Wyman [375]: I
Bn
r+1
en(r+l/r-I)-1
(7.9)
The following theorem is due (in a slightly different form) to Harper [256]. In the proof we follow Canfield [92] using Hayman's method.
be defined by PQn = k) =
Theorem 7.1.9. Let the integral random variable
Let µn := er - 1 and Qn := r+l
-
1 = An - r+l, where r is the root of
re' = n. Then n is asymptotically normal with mean An and variance
Proof. Let pn (y) := Ek o Sn,kyk. Then the characteristic function of 4n is given by cps,, = pn(eit)/pn(1). Moreover, to(5n-l2n)/Qn(t) =
e-'
"QnPn(e't/an)/pn(1)
(7.10)
By Theorem 7.1.1 it is sufficient to prove that the RHS tends to e_t2/2 if n -+ oo. Thus we will determine an asymptotic formula for p n (e't /°n ). In view of Lemma 7.1.1 00
Zn
E Pn (Y)
. II
=
ey(e-- I )
n=0
which implies eY(e`-1)
Pn (Y)
n!
27ri
Izj=r
zn+1
dz = r
n
jr
eY(ere
-1)-ntcpdcp.
27r ,/_n
As in the proof of Stirling's formula, we write the integral on the RHS as a sum I1 + 12, where II is the integral over f cp l < 8(n) =: 8 and 12 is the integral over 8 < I tp1 < ir. First we investigate II. From (7.10) we know that we must later put y := e't/°ry (resp. y := 1); that is, in both cases y := e's where Isi is small (s = t/Un or s = 0). Hence we expand also e'5 (at least for members of higher order) in the Taylor expansion of the exponent of the integrand g(s, (P) :=
e's(er,"' - 1) - nice.
7.1 Central and local limit theorems
315
We have g(s, 0) = e's(er - 1), a
app g(s, 0) = e`sirer - in = i(rer - n) - srer + rer O (s2).
We determine r such that the imaginary constant term vanishes; that is, rer = n. Then a
apg(s, 0) = -sn + O(ns2), a2
Wg(s, 0) = -e`ser(r2 + r) = -n(r + 1) + O(nrs), a3
,
g(s, (p) = -ie'sere (r3 a3'9 + 3r2 a2'' + re"') = O(err)3 = O(r 2n).
We will take a S such that IsI < S for large n. This is clear for s = 0. For s = a this inequality is satisfied if 1) -+ oo, that is, S2r -+ 00. Condition 0. S2(r+l+l Moreover we have k I < S. Thus Taylor's formula gives
-
2
g(s, cp) = eis(er - 1) - snrp - n(r + 1) 2 + O(nr233)
= eis(er - 1) +
s2n2
2n(r + 1)
_
(
sn/ n(r + 1))2
n(r +
2
The substitution tr := n (r + 1)cp + sn / n (r + 1) yields the bounds of integration n(r + 1)(±8 + r+l+l ), which are in absolute value not less than n -(r+ 1)8 (1 - ). If we integrate from -oo to oo we obtain the value
r eei's(er-1)+szn/2(r+1)
1
n(r+1)
2n,
and this is the asymptotic value for Il uniformly for Isl < 8 if Condition 1. n(r + 1)8(1 --+ oo; that is, In-r8 -> oo, and (omission r) of O(nr283)) Condition 2. nr283 -+ 0. Again we want to make 12 small. We have 12 = eg(s,O)
e9(s,w)-g(s,o)dW.
1 sj
< w(N3) + Y, Vw(N;) + E Aw(N;). i>j
i<j
Also, if h > j, in view of the induction hypothesis and (7.14) (note that A w (q *) _
w(q*))
d(Q, w) = d(Q', w') Aw'(N; - {q*}) < w'(Nj - {q*}) + 57, Vw'(N1) + i<j i>j
< w(N3) + EVw(N;) + E Aw(N1) - w(q*) i>j
i<j
+
g(q*q)w(q*) q:q>q*
< w(Nj)+EVw(N;)+EAw(N;) i<j
i>j
since Eq:q>q* g(q*q) < 1 because of (7.11).
The following theorem is in the case v - 1 due to Alekseev [23]. We proved the general case [152]. Theorem 7.2.1 (Asymptotic Product Theorem). Let (P, v) be any positively weighted poset and assume that P is not an antichain. Then
d((P v)") where v(P) :=
(v(1'))" asn -->oo, 2nno,(P, v)
v(p) and v2(P, v) is the variance of (P, v).
Proof. First observe that we may make the general supposition that v(P) = 1 (i.e., the weight function v is normalized) because d((P, cv)n) = cnd((P, v)n) for any positive constant c, n = 1, 2, .... Let x be an optimal representation of (P, v) with Ax = 0 and define y : P" -> R by y(p) :_ yi_1 x(p;), where p = (pl, ... , pn) E P" (by Theorem 4.6.6, y is an optimal representation of (P, v)n). To get a lower bound we take in (P, v)n the antichain A := Ay(0) as defined at the beginning of this section. We consider the discrete random variable
defined by P(r; = a) = v({p E P : x(p) = a}). The expected value it of
Limit theorems
320
equals it, = 0, and the variance a2 of equals aS = a2(P, v). Let i, ... , " be independent copies of and let n := + + ". Then
(v x ... x v)(A) = PH-2 < n < 2 If has a nonlattice distribution, we can derive directly from Theorem 7.1.6 that 1
Ji27rna(P, v)
asn -)- oo.
(7.15)
If has a lattice distribution, the same follows from Corollary 7.1.2 if the maximal span h of l; has the form h = 1 for some natural number j. Since P is not an antichain and x is an optimal representation of (P, v), there must be two elements an integral p, p' E P such that x(p') - x(p) = 1. Because, by definition, random variable for some a E R, we have for some natural number N
£ is
x(p') - a
x(p) - a
- N,
h = N.
i.e.,
Thus (7.15) is also proved in the case of a lattice distribution and we infer
d((P,
v)') >
1
2nna(P, v)
It remains to show that
d((P, v)")
r) < 2e-Iog2n/(8vi(I-vi)) = o
1
I
J.
Hence,
Vw(q) + E Ow(q) < E w(q) = w(Q - G) = o (k). q:gVG: z(q) proved analogously). Then
323
(the corresponding inequality for Vw(q) can be will be complete because (7.24) yields
the2proof
E Vw(q) + E Ow(q)
E w(q)o ( V ) _ o I
q:qEG
qEG:
qEG:
z(q)2
(1V)
x v) (P") = (V (p)), = 1 by the general supposition). (note that w(Q) = (v x Let q E G, z(q) > 2. If Ow(q) = 0, then (7.24) is trivially true; thus let tw(q) > 0. Then
Ow(q) = w(q) 1 -
(7.25)
w(q) ) Here and in the following, the sums are extended over all q' such that q' 0, pi 0 are satisfied automatically for sufficiently large n since q E G. Hence we have equality in (7.19):
E g(q'q) = 1. We have (for q'=(qt,..., qi+1, ..., qj-1,...,qk)) w(q') _ w(q)
(7.26)
yigj vj(gi + 1) vi
I
qi + 1 + (qt + 1)n
(vj
vi) (n
vi
vi
1 (qj _qi +n vj vi It can be easily checked that, uniformly in q E G, 1
qj + 1 Vi
qj
(qi + 1)n
vj
_
qj
vi
n --v;
0 for all q with z(q) > 2 .
,
Limit theorems
324
Since f is a representation flow on (Px, v) we have by definition of g
g(4'4)
qJ J
- 9i /
= u Ff(P jPJ) \JvJ F
t-t
i (Par vi
vi / qj
- Pi) t _ F
vi
vixi = -z(q) > 0. F
Thus not only the proof of (7.24), but also the proof of (7.16) and the proof of the theorem are complete.
In fact, in the case v =- 1, Alekseev used this result to derive an asymptotic formula for the number ipp,Q(n) of order-preserving maps from Pn into Q: Let SQ be the (finite) set of all finite sequences (Ho, H1, ... , H,s) of subsets of Q such
that IHol=1,HI=1andforalli=0,...,s-1,pEHi,gEHi+l we have p
q. Let r(Q) := max{IHoIIH1I ... IHs I : (Ho, ... , Hs) E SQ}.
Theorem 7.2.2 (Alekseev [23]). If P and Q are not antichains, then IPIn
wPP,Q = T (Q) 2nna(P)
O+oO)
In the proof, a method of Kleitman [301] is used. We will omit the proof but construct as many functions as given in the theorem: Let r(Q) be attained by the sequence ({qj}, H1, ... , H5_1, {q1,}) and let x be an optimal representation of P = (P, 1). We take again the antichains
Ax(i):={pEP':i-2 <x(P)s-2}.
Ax (> s-1)
Obviously, any function that maps Ax(< 0) onto {qj}, Ax(i) in any way into Hi, i = 1, ... , s - 1, and Ax (> 0) onto {qu}, is order preserving. Consequently, IHI I IAx(1)I
(PP,Q
... IHs-1
IIAx(s-1)I.
Exactly as in the proof of Theorem 7.2.1, we may derive from Corollary 7.1.2 that n
Ax(i)I=
IPI
27rna(P)
(1+0(1)) for every i=l,...,s-1.
Consequently, (PP,Q(n) ? (IH1I ... IHS_I I)
2an o(P)
(I+o(t))
2nnPin
= r(Q)
a (P) (I+o(1))
325
7.2 Optimal representations
Also without proof we mention another variant of Theorem 7.2.1 which we found together with Kuzyurin [163]:
Theorem 7.2.3. Let { P" } be a sequence of posets (v =_ 1) that are not antichains
and whose sizes are bounded by some fixed constant. If k = k(n) = o(J), then
dk(Pl x
kIP1I
x P") ^
I P" I
as n --> oo.
21r n-1 U2(P,) For a ranked and weighted poset (P, w), let
b(P, w) := max w(N1). t
Proposition 7.2.2. If r(P) > 0 then
b((P _)n) ti (w(P))" asn -moo ,
v'2-nna,.
where oY is the variance of the rank function.
Proof. Apply Theorem 7.1.5 to the independent random variables j, j = 1, ... n, where P(t; j = i) = W(P) i = 0, ... , r(P). Note that no; is the variance of the rank function of (P, w)". Moreover, recall that we proved in Corollary 7.1.3 for the partition lattice II" with w - 1 that also
b(IIn)
Innl 21rQn
Bn
2ncrf
(7.29)
In contrast to the usual notation, we write Q f instead of Q, in order to avoid confusion with the number r (the solution of re' = n). Here we may replace or, by o,- f because of Remark 7.1.1.
Let {(P,,, wn)} be a sequence of ranked and weighted posets. We say that (Pn, wn) has the asymptotic Sperner property if d(Pn, w,,)
b(P,, wn)
as
n -* oo.
Corollary7.2.1. Let (P, w) be a ranked and positively weighted poset. (P, w) is rank compressed iff (P, w)" has the asymptotic Sperner property.
Proof. The case r(P) = 0 is trivial; for r(P) > 0 apply Theorem 7.2.1 and Proposition 7.2.2.
Note that not all rank-compressed posets have the Sperner property and that the strong Sperner property does not imply rank compression; see Figure 7.1.
Limit theorems
326
Figure 7.1
The question whether the partition lattice IIn is asymptotically Sperner or rank compressed had been open for a long time. Finally Canfield and Harper [98] found that the answer in both cases is no.
Theorem 7.2.4 (Canfield and Harper).
d(IIn) > n1/35 b(nn)
Proof. Let r
We have for sufficiently large n
Q(nn) < Qrf(nn)
and
1 1
335
and r be the solution of rer = n. We define the sequence
by
if [rrJ < j < [2trJ, otherwise.
Then we introduce the function x : nn - R by x(7r) := A(n) B11
+
+ A(n) I& if 7r En, consists of the blocks B1, ... , Bk.
Note that if Ann) was proportional to all j, then x(7r) would be constant; that is, it would have variance 0. By checking several cases, one finds that x is a representation of IIn (the dual of IIu). The large interval in which Ann) is proportional to j yields a sufficiently small variance. If we define Zn := {x(7r) : 7r E 1-ln} and
an,a := I{7r E 11n : x(7r) = all, then we may prove analogously to Lemma 7.1.1 that 00
e_j=1 Y A
JD
[
=k=0 E UeZ L ann!
),p Zn.
By Theorem 7.1.11 (noting 2 < vo), the random variable cn, where a) = I{7r E IIn : x(7r) = a)I/Bn, is asymptotically normal with mean µn and variance on given in Theorem 7.1.11. Using Remark 7.1.1, we may calculate V
(which is the variance arx2 of the representation x) with high precision: Let
7.2 Optimal representations
327
J := Lrrj, K := L2rrj. Then 00
(n)) 2 r
j
j=l
j
j! 00 (12
00 j2
1
(1 - r)2
(
(1
r2 j1 +
rJ
J
r)2
- r2)
J
r2 - 1)
j!
j=K
rJ
,2
j!
rJ
/
\\1+r/er+O\J+O\K!//
(1
since the second and third sum are bounded by geometric series whose ratios are bounded away from 1. In a similar manner, 00 1
j=l
r ((r
Ki I+r0()).
+1)er+rOl
\ /
./
Consequently,
x\
0 jl + Kl l + r 0 ((Jl /
V
x
Kl)2) + O(1).
+
n \ rJ 4er/4 rK ) (it was the By Stirling's formula we find that both J! and K, are of order O ( -117 aim to attain the same order during the determination of r). Thus,
r(2? l
rJ
K
er/2
(fl(elo2)/2
(e (re log 2)/2
rs
where := 11+e2 g 2 (we replaced er by 11). Hence we are also able to express the r variance as a power of n: n (e log 2)/2
r
By Theorem 7.1.9 and Remark 7.1.1, o f(IIf) ^
,
which yields
2
= rf
n
./35
forn large enough.
(7.30)
//
Accordingly, the second inequality in the statement of the theorem is proved. To verify the first inequality, we apply Chebyshev's inequality: We have (with
ox = V(W) P06, - EQn) I < tax) > y. The interval (E(on) - 2Qx, E(on) + 2ox) can be covered by disjoint half open intervals of length < 1, using at most 40x + 1 such intervals. Hence there exists
Limit theorems
328
at least one interval I = [a - 2, a + 2) such that 3
4
4vx+1' and Proposition 7.2.1 yields 3
4
d(nn) > Bn
4ax+1
With (7.29) and (7.30) we derive for some constant c > 0 d(IIn)
b(nn)
>c>n Qrf
1/35
cr
Wp) can be bounded from above in the following For arbitrary posets, the ratio d(P) way (we omit the proof, see [164]).
Theorem 7.2.5. Let P be a ranked poset of size k. Then (a)
d (P)
1 [k+2
b(P)
2
'
2 kk+2
(b) lim n-*oo
if k is even,
4
d(P")
0 let where or depends an k and A, only. Let po := 1/
N11,E(n, k) := {x E Ni(n, k) : z(x) V [(po - E)n, (po +E)n]}, Wi,E (n, k) := I Ni,E (n, k) I
Claim 1. Let 0 < E < po(1 - po). Then there exists some constant c such that Wi,E (n, k)/ Wi (n, k) < e-`n, that is, the number of elements x of Ni (n, k) containing fewer than (p0 - E)n or more than (pp + E)n zero coordinates is exponentially small in n with respect to Wi (k, n).
Proof of Claim 1. Let n be a random variable with P(ro) = 1 - po and P (rl = 1) = po. Let rj i ... , rin be independent copies of ri and let wn : = n i + + rln. Then P(wn - h) -
(n)
h(n) (a + a2 + Ph(1
kknn
- po)h
h(1 + a +
+
ak)n-h
+ ak)n
aj
0(l+a+
+ak)IIXENj (n,k):z(x)=h}I.
Limit theorems
330
Furthermore, in view of Theorem 7.1.3, for some constant d > 0,
E
2e-dn > p(I pon - con I > En) _
P(wn = h)
h¢[(p0-E)n,(p0+E)n]
aj
krn
= j=0 ` (l + a +
I {x E Nj (n, k) z(x) = h}I
+ ak)" hV[(po-E)n,(po+E)n]
a' Wi (n' k)' > (1 + a + ... +. ak)n With (7.32) we obtain, for some constant 0 < c < d W1,e(n, k) < e-cn
W; (n, k) Claim 2. IF[r]I > W;(n,k)/(r) if i > kt. Proof of Claim 2. If i > kt then I supp(x) I > t for all x E N1(n, k). Thus Ni (n, k) = UFX, implying W; (n, k) < are extended over all X E
(n)
I Fx I =
I F[r] I (the union and the sum El
(a) Let k < Ai . Then, by (7.31), a < fir and, in view of the definition of ,Br, +01 k < 1. Consequently, po > t+-L1 . Let 0 < e < min{po(1 - po),
a + a2 +
po - t+1 } Finally let F' := F - N1,E(n, k) and F,r]
F[r] - N,,E(n, k). Since
by Claim 1 and 2 WI,E(n, k) < Wi,E(n, k)
1 uniformly in + 3 < 1 < 1 - S. n *oo (n-t) t+1
Thus there exists some constant c > 1 such that ICI, jI > for j E [(1 po - E)n, (1 - po + E)n] and n large enough. Consequently, for these n, IFI > IF*I > cJF,t]l - cIF[t]I. (c) Let ,l > X*. Analogously to (a) we obtain po < 2 , and we find thus some 8, E > 0 such that z - 8 > po + E and e < po(1 - po). We put F* := {x E N; (n, k) : I supp(x) I > (2 + 6)n}.
Then, for sufficiently large n, F* is a statically t-intersecting family because for x, y E F* it follows that I supp(x) n supp(y) I > 28n > t. By Claim 1, 0
IF*I.W;(k,n).
61
Macaulay posets
In this final chapter a theory is presented that is based on the Kruskal-Katona Theorem (Theorem 2.3.6) and its predecessor, the Macaulay Theorem (Corollary 8.1.1). The central objects are the Macaulay posets, and the main theorem says that chain and star products are Macaulay posets. These theorems provide solutions of the shadow-minimization problem, but several other existence and optimization problems can be solved in this theory as well. We restrict ourselves to chain and star products (and their duals) because these have many applications, and both posets are very natural generalizations of the Boolean lattice. Recall that we already know much about these special posets S(ki, ..., and T(kl, ..., We proved that they are normal and have log-concave Whitney numbers; see Example 4.6.1. In particular, they have the strong Sperner property. Furthermore, chain products are symmetric chain orders; see Example 5.1.1 (and, moreover, ssc-orders; see Example 5.3.1), and are unitary Peck; see Example 6.2.1. A more detailed study of isoperimetric problems, also in other structures, such as toroidal grids, de Bruijn graphs, and so forth, will be presented in the forthcoming book of Harper and Chavez [261]. We refer also to Bezrukov [57, 58] and Ahlswede, Cai, Danh, Daykin, Khachatrian, and Thu [6]. In the case of chain products, we will further investigate two types of intersecting (resp. cointersecting) Sperner families and discuss some other properties. A complete table for several classes of families, as it was given in Chapter 3 for the Boolean lattice B, is not known and seems to be difficult to find since Katona's circle method does not work here. However, in some cases the succeeding theory may replace the circle method. Further results for families in chain products that satisfy certain conditions but are not Sperner families can be found, for example, in [161].
332
8.1 Macaulay posets and shadow minimization
333
8.1. Macaulay posets and shadow minimization Let P be a ranked poset and -< a new linear order on P. Clearly, -< induces also a
linear order on each level N; For a subset F of P and a number m, let C(m, F) (resp. £(m, F)) be the set of the first (resp. last) m elements of F with respect to - 1.
(8.1)
Though there may exist several linear orders such that (8.1) holds, we assume in the following that with a Macaulay poset always some fixed linear order < satisfying (8.1) is given; that is, we consider Macaulay posets as triples (P, >k
for S(kl,... , k,) and kl < ... < k, for T (kl, ..., kn). For the levels, we use sometimes the notation Ni ( k 1 .
.
. . .
kn) to indicate the parameters (from the context
it will be clear whether chain products or star products or both are considered). Note that the rank of S(k1, ... , kn) is not n, but s := kl + + k,,. First we need
the corresponding linear orders 1. Let {b E N, : bn < an },
F1
F2 _ {(b1,
, bn-1) E Ni-an (kl,... , k1)
:
(b1, ... , bn-1, an) E F).
Since F is compressed, we have F1 C F. Moreover, F2 is compressed, too. Obviously,
Li(Fi) _ {C E Ni_1 : cn < an} and
A(F) = O(F1) U {(c1, ... , cn-1, an) : (ci,... , cn-1) E A(F2)1. By the induction hypothesis, O(F2) is compressed; thus A(F) is also compressed. Now we consider T (kl , ... , kn ). For any vector b and any number 1, let b T I be the vector that can be obtained from b by deleting all components equal to 1. Let 1 := min{aj : j E [n]}. Let
Fl :={bEN;:b(j)#0for some jE{0,...,1-1)) U{bEN1:a(1) k2 - k1 + 1, then (k1 -,8') - (k2 - a') E {0, 1). A product with integral factors of constant sum attains its maximum if the factors are almost equal; thus (8.11) is proved also in this case.
Now we prove (8.2) by induction on n. The cases n = 1, 2 are settled. So let us look at the step n - 1 n > 3. We still need some preparations. For F C Ni, we put Fj:d
{(al,...,an) E F: aj =d},
Fjjd
{(al,...,aj-1,aj+l,...,an) E Fj:d}.
Let, forS(kl,...,kn) andd> 1,
Aj:d(Fj:d) := {(al,...,d - 1,...,an): (at,...,d,...,an) E Fj:d} and, for T (kl, .
.
.
, kn) and d = k,
Oj:d(Fj:d) := {(al,...,e,...,an) : (al,...,d,...,a.) E Fj:d,
kn-kj<e 1,
for T(kl, ... , kn),
d=k
for
Ukn-ki<e 1,
Ukn-kj <e 1,
and forT(kl,...,kn) O(Cj:dFj:d) C
Cj:dGj:d
ifd 1, we have ej = k and i = 1. Let a, = k, r
li(F')I = lo((F - {e}) U {e))I _ Ji(F - {e})I + li(e) - i(F - {e})j, li(F*)I = l i((F - {e}) U {a})I = Ji(F - {e})I + li(a) - i(F - {e})j. Thus, in order to prove (8.16), we only must verify that
Ii(a) - i(F - {e})I < Ii(e) - i(F - {e})I.
(8.17)
Let µ(x) := max{xh : h E [n] - {r, j}}.
Claim 6. We have Ii(a) - i(F - {e})I < k - µ(a). (al, ... , ar_1, a, ar+l, ... . a E {0, ... , µ(a) - 1}. Let a,s = µ(a) (note s 0 r), and
Proof of Claim 6. It is enough to show that a' define a* by
ifh=s, if h = r, otherwise.
Since a and a* coincide in all but two components and a < µ(a), it follows that a* -< a. By the choice of a, a* E F - {e}. Consequently, a' E i(a*) c
i(F - {e}).
Claim 7.Wehave Iz(e)-i(F-{e})l >k,,-µ(e)-1. Proof of Claim 7. It is enough to show that e' := (el, ..., ej_1, e, ej+l, .. .
e (= {µ(e) + 1, ... , k - 1). Thus assume that e' E i(e*) for some e * E F - { e } where
k ifh=s, =
e
ifh = j,
eh
otherwise.
Since e and e* coincide in all but two components and e > µ(e), it follows that e -< e*. By the choice of e, e* V F, a contradiction. By Claim 6 and Claim 7, (8.17) is proved if we can show that
k,, - µ(e) - 1 >- k, - g (a), that is,
E.c(a) > fc(e) + 1.
8.1 Macaulay posets and shadow minimization
343
Assume the contrary. Then there must be some index s E [n] - jr, j} such that
es=ti (e)> s(a)>as. Let a be that element which can be obtained from e by replacing the sth component
es by as. Then clearly e -< e, and since F is j-compressed, e E F. Obviously, e (0) = a (0), ... , e (1- 1) = a (1- 1). Moreover, it is easy to verify that i(l) Wh < Wh+l . Let F2 := Nh and F1 C Nh+l such that IF1 I = IF2I. Then, using that P is graded, O(CF2) =
Nh_1 and O(CF1) C Nh. Consequently, sfh(F2) = Wh_1 > Wh ? sfh+l (Fl), a contradiction. Now the asserted inequality follows from an iterated application of (8.24).
The following theorem is for S due to Clements [106] and for T and Col due to Leck [334]. Theorem 8.1.3. The Macaulay posets S(kl, ... , k,), Col (kl , ... , k,), and T (kl , .... kn) are shadow increasing.
Proof. We again study the posets separately.
Case 1. P = S(kl, ... , kn ). We put Q := S(kl, ..., kn, 1). For Fl C Ni+l (P) and F2 C N1(P), we define
G1 :_ {(a, 0) : a E CFl},
G2 :_ {(a, 1) : a E CF2}.
Macaulay posets
350
Clearly, G 1, G2 c N1+i (Q), and both families are segments where G, is initial. Obviously,
Onew(G1) _ {(b, 0) : b c 0(C Fl)}, Onew(G2) _ {(b, 1) : b E 0(CF2)}. Theorem 8.1.2 and Proposition 8.1.4 imply for I Fl I= I F21, that is, I G 1 I= I G21,
sj (F2) = IA(CF2)I = Ionew(G2)I
Ionew(G1)I = Ii(CFi)l = sJ+1(Fi).
Case 2. P = T(kl,... , kn). We put Q := T(kl,... , kn, kn). For F1
C_
Ni+1(P) and F2 C Ni (P) with I Fl I = I F21, we define Go Gkn
{(a, 0) : a E CFI), {(a, kn) : a E CF2}.
Then Go, Gkn c N,+1 (Q). Let (e, kn) be the last element of Gkn with respect to
.Let, fori=1,...,kn-1, G, G1
(e, kn)},
(a E P : (a, i) E G,),
G := Ukn 1 Gi. It is easy to see that each G;, i = 1, ... , kn - 1, is compressed, that G is a segment in N,+1(Q), and that
Onew(G) ? {(b, k,,) : b E A(CF2)} U (Uk° i'{(b, i) : b E 0(G;)}) .
(8.25)
Note that G has size Fl I + F k " i' 1 G, I . W e partition CG into consecutive segments Ho__ , H k n _ 1 such that I H o I = 1 F l l and I H i I = I G i l , i = 1, ... , kn - 1. Then CHo = Go and, for i = 1, ... , kn - 1, CHi = ((a, 0) : a E G'). Consequently,
IO(CHo)I = lo(CFi)I,
I A(CHi)l = lo(G;)l,
i = 1, ... , kn - 1. (8.26)
From (8.25), (8.26), Theorem 8.1.2, and Proposition 8.1.4 we derive kn-1
Ii(CFi)l +
kn-1
Ik(G;)l = E Io(CHi)I i=1
i=0
lo(Uk"o'H1)I = Ii(CG)I kn-1
Iz(G;)l
lonew(G)l >- 1o(CF2)I + i=1
8.2 Existence theorems for Macaulay posets
351
and finally
Sf+1(F) = IA(CF1)I >_ IA(CF2)I = sf (F2)
Case 3. P = Co1(kl, ... , kn ). We put Q := Co1(kl, ... , kn , kn + 1), that is, kn+1 := kn + 1. Let t/r : [nkn]
[(n + 1)(kn + 1)] be defined by
i/i(gn+r):=q(n+l)+r (rE[n]). For X c [nkn], let '(X) := {fi(x) : x E X}. It is easy to verify that X E P implies i(X) E Q. Thus For a family F, let >r(F)
can be also considered as a function 1 : P - Q. {i(X) : X E F}. Obviously, for F C N1(P),
ik(0(F)) = o(*(F)) For F1 c N;+1(P), F2 c Ni (P), we define G1 :_ {,/f(X) : X E CF1}, G2 := {t/r(X) U {(n + 1)kn+1} : X E CF2}. Then G1, G2 c N1+1(Q) and G2 is a segment. Obviously, Lnew(G2) = {Y U {(n + 1)kn+1} : Y E o(tp'(CF2))}. Theorems 8.1.1, 8.1.2, and Proposition 8.1.4 imply for I Fl I = 1 F21, that is, I G 1 I = IG21,
Sf (F2) = Io(CF2)1= Io( (CF2))I = Ionew(G2)I < Ionew(CG1)I
= Io(CG1)I < It (G1)I = Io(1(CFI))I = IA(CF)1 = s f.+l(F1)
8.2. Existence theorems for Macaulay posets In this section we will mainly characterize the profiles of ideals and antichains in Macaulay posets. Recall that
F={pEF:r(p)=i}, f=1F11 We say that a family F c P is compressed if CF = F, for all i. Obviously,
F is an ideal if 0(F;) C F_1 for all i > 1.
(8.27)
For a natural number 1, let E , (1) be the value of the shadow function of an 1-element
family in Ni, that is,
Ai(l) := IO(C(1, N1))I
Macaulay posets
352
Theorem 8.2.1. Let P be a Macaulay poset. The following conditions are equivalent:
(i) f is the profile of an ideal in P, (ii) f is the profile of a compressed ideal in P,
(iii) A; (j) < f_1 for all i > 1. Proof. (i) -+ (ii). Let F be an ideal with profile f. Let G := UjCF1. Then G is compressed and has also the profile f. Moreover, by (8.1),
A(Gr) = A(C1) C C(A(F)) S CF-1 = Gi-1, i > 1; thus, in view of (8.27), G is an ideal. (ii) H (iii). Let F = N1). Then F is compressed, and F is by (8.27)
an ideal if
A(C(fi,N,)) cC(fi-1,Ni_1)foralli > 1. Since by Proposition (8.1.1) the shadow of an initial segment is an initial segment, the last inclusion is equivalent to (iii). (ii) -> (i) is trivial.
Now we will derive a generalization of the equivalence (i) H (ii). Let (P1, (ii). We construct for a given Sperner family F with profile f the canonically compressed Sperner family G with profile f as follows: We define
Gn := C(fn, Na),
and fori=n-l,n-2,... Gi := C(fi, N1 - A .i(U;=i+1Gj))
(8.28)
(soon it will be clear that the construction works). Claim. For all i = n, n -1, ... for which the construction (8.28) is still possible (i.e., f < I N i l - I .i (U 1+1 G j) I ), there holds
A,;(Unj=i+1Gj) is compressed, (a) H, (b) Hi c CA-.i (U j=i+1 Fj ). Proof of Claim. We proceed by induction on i = n, n - 1, .... The case i = n is trivial (note Hn = 0). Thus look at the step i + 1 -+ i. (a) By the induction hypothesis and construction, Hi+1 U Gi+t is compressed.
Since O.+ (Gj) = 0(O,i+1(Gj)) for j > i, it follows that Hi = i(Hi+l U G1+1) and because of Proposition 8.1.1, Hi is compressed. (b) A-.i+1(U j 2Fj) fl Fi+1 = 0 since F is a Sperner family. Together with the induction hypothesis this provides
.
Hi+l U Gi+1 9 C(&-.i+1 (U j=i+1 Fj)).
8.2 Existence theorems for Macaulay posets
355
Thus,
Hi = A(Hi+1 U Gi+l) C A(C(A i+l (U' C C(A(A
1 F'i)))
+l (U j=i+I Fj)) = C(0->i (U j=i+1 Fj)).
Here the second inclusion follows from the fact that P is a Macaulay poset.
As already mentioned, F, fl A,i
Fj) = 0; hence
f 5 INil-A,j(U=i+,F'j)I:5 1N11 - Phil. This shows that Gi can really be constructed for all i > 0. By construction, G is a Sperner family, and by (a) of the claim, O,i (G) = Hi U Gi is compressed for all i > 0.
(ii) -* (iii). Induction on i = n, n - I.... easily yields that the LHS equals A-4i (F) I, where F is the canonically compressed Sperner family with profile f.
(iii) - (ii). The condition says that the construction (8.28) is possible up to
i=0. (ii) - (i) is trivial. We use the notation CF for the canonically compressed Sperner family which has the same profile as F. From the proof of Theorem 8.2.3 (in particular Claim (b)) it follows easily that
A,i(CF) c C(A i (F)) for all i.
(8.29)
Without going into details we mention some applications in polyhedral combina-
torics. Theorem 8.2.1 gives in the case P = B a characterization of f-vectors (profiles) of simplicial complexes (Kruskal [325], Katona [292]) and of polyhe-
dral complexes (Wegner [460]). In the case P = S(oo,... , oc), Theorem 8.2.1 together with Theorem 8.1.1 permits a characterization of the f -vectors of Cohen-
Macaulay complexes (Stanley [438]), and in the case P = Col (d, ... , d), these theorems allow a characterization of the f-vectors of (d - 1)-dimensional completely balanced Cohen-Macaulay complexes (Frankl, Fiiredi, and Kalai [197]). A structural characterization of the f-vectors of balanced Cohen-Macaulay complexes is given by Theorem 8.2.2 applied to Pi = S(oo,... , oo), where the number of components may differ (Bjorner, Frankl, and Stanley [66]). Finally, in the case P = B, Theorem 8.2.3 provides a characterization of Betti sequences over some field of some simplicial complex and of some polyhedral complex on at most n + 1 vertices (Bjorner and Kalai [67]). We refer the reader to the surveys of Bjorner [63, 65], Bjorner and Kalai [68], and to the books of Stanley [444] and Ziegler [473].
Macaulay posets
356
8.3. Optimization problems for Macaulay posets For F C P, q E P, we write in the following F -< q (resp. q -< F) if p -< q (resp. q -< p) for all p E F. We call a Macaulay poset P rank greedy if the associated linear order - is a linear extension of the ordering < of P - that is, if
p < q implies p : q, and if
A(P) -< q, r(p) > r(q) imply p -< q.
(8.30)
The motivation for this notion comes from the construction of linear extensions. Suppose that we have constructed already a set F of first several elements. The
next element p E P - F must have the property A(p) C F. From all elements with this property we take one with largest rank as the next element.
Proposition 8.3.1. If Pisa rank-greedy Macaulay poset, then so is its dual P*.
Proof. As in the proof of Proposition 8.1.2 we take the dual associated linear order for P*. Since - is a linear extension of
of < as the trivially also
- r(a). Let E be the ideal generated by
e. Note that E C F*. For all f E E with r(f) = r(a), we have f -< a since otherwise a E F* (recall that F* is compressed). Let e* be a minimal element of E with the property a e*. By the preceding remarks, r(e*) > r(a). The choice of e* yields A(e*) -< a. By (8.30), e* -< a, a contradiction. Several authors contributed to this theorem. Ahlswede and Katona [13] studied Bn and considered also other types of weight functions. Bezrukov and Voronin [60]
then gave a generalization to S(kl, ..., k,). After preparatory work of Kruskal [326] and Lindstrom [345], Bezrukov [56] settled T (k, ... , k). He finally also proved Theorem 8.3.1 in an equivalent formulation [57]. Earlier special weight functions like w (i) = i have been considered (e.g. for S(kl, . . . , kn)) by Lindstrom and Zetterstrom [347] (k1 = ... = kn) and Clements and Lindstrom [117]). This theorem provides a solution of the maximum-edge problem for certain graphs; see p. 40. First let G be the Hamming graph of S(kl, ... , k,); that is, the vertex set V of G equals S(kl, ... , kn ), and we have a b E E iff I {i : ai 0 bi) I = 1.
Recall that for Fc V,E(F):={eE E:ec F). Theorem 8.3.2.
For the Hamming graph of S(ki, ... , k,) and for 0 < m
... > I F(n(kn))I We have
k IE(F)I = L IE(F(i))I + i=0
I{ab E E : a E F(i), b E F(j)}I 0 ... > I G (kn) I ; that is (a,, ..., an-1, i + 1) E G implies also (a 1, ... , an_ 1, i) E G. Consequently, G is an ideal. For any ideal G, I E(G) I can be written also in the form
IE(G)I = E I{b E G : b < a, ab E E(G)}I = E r(a). aeG
aEG
Finally, by Theorem 8.3.1 and Proposition 8.3.2, we have I E(G) I
E(C(m, S(kt ,
.... kn)))I. Since the Hamming graph of S(kl, ... , kn) is regular of degree ki + + kn, Theorem 8.3.2 provides also a solution of the edge-isoperimetric problem; see p. 40. This is a result of Lindsey [343] (see also Clements [103] and Kleitman, Krieger, and Rothschild [307]).
Macaulay posets
360
Now let G be the Hasse graph of T (kl , ... , kn ), which is the same as the Hasse graph of Col (kl , . . . , kn ) . I prefer the formulation with T (kl, ... , kn) because of a succeeding application. Thus the vertex set of G equals T (kl, . . . , kn ), and we
have ab E E if for all but one i, a, = bi, and for the exceptional i, ai = kn and bi # kn or ai # kn and bi = kn . We have in the case of the Hasse graph of T (kl, ... , k,) for
Theorem 8.3.3.
0<m (p (b), then a < b.
(b) If F is an ideal in S(kl,... , kn), then cp(F) is a filter in T (kn, ... , k1). (c) If a < b then cp(b) < rp(a) (where -< is given by (8.8)).
(d) tp-1(G(m, T (kn, ..., k1))) is an ideal in S(k1, ..., kn) for each m. Further, we will need the following lemma:
Lemma 8.3.2. Let ao, . . . , ak be real numbers and let 7r be a permutation of {0, ... , k} such that a,r(o) > > a,r(k). Then k
k
L min{ai_1, a; } < 5 min{a,r(i_1), a,r(q}. i=1 i=l
Proof. The RHS equals Ek=1 a,(,). Let s; := min{ai_1, ail, i = 1, ..., k, and let or be a permutation of 11, ... , k} such that sa(1) > > Sa(k). We only must show that s, (j) < a, (j) holds for j = 1, ... , k. Assume the contrary, that is,
sa(1) > ... > So(j) > a,(j) for some j. In the following we consider the sets as multisets; that is, elements may appear repeatedly. Let Si := Jai-1, ail, i = 1, ... , k. By our assumption, all elements of Ui SS(i) are greater than a,r(j). Since a,r(j) > a,r(j+l) > , it follows that U1 SQ(i) c {a,r(o), ..., a,r(j_1)}. However, Ui=1 Si clearly contains at least j + 1 elements, a contradiction. Theorem 8.3.4. For the Hasse graph of S(k1,... , kn) and for 0 < m < (k1 + 1)
(kn+1),wehave max{IE(F)I : F C S(kl, ... , kn), I FI = m}
= IE(1P-1(1(m,T(kn,...,kt))))I Proof. We again use the approach and the notations from the proof of Theorem 8.3.2. Here we obtain k
kn
IE(F)I L IE(F'(i))I +min[ IF'(i - 1)1, IF'(i)I}. ;=l
i=o
Let n be a permutation of {0, .
. .
, kn } for which
IF(n(0))I >_ ... > IF(ir(kn))I
Macaulay posets
362
We define the new family G by
G'(i) := W-1(G(I F'(n(i)) I , T (kn-1, ... , kl))) Then
k
k
IE(G)I = ' I E(G'(i))I + L min{1G'(i -1)I, I G'(i) I }. i-o i=1 By the induction hypothesis and Lemma 8.3.2, IE(G)I > IE(F)I. By Lemma 8.3.1(d) and construction, G is an ideal in S(k1,... , kn). We have for any ideal G in S(k1, . . . , k,) in view of Lemma 8.3.1(a) and (b), n
I{b EG:b 0}I, aEG
1] I{i E [n] : ai < kl}I = IE(gP(G))I aEr7(G)
From Theorem 8.3.3 we derive (noting Lemma 8.3.1(d))
IE(G)I = I E((P(G))I < I E(.C(m, T(kn, ... , kl)))I
= I E((P-1(G(m, T(kn,... , ki))))I.
In the case kl = . . . = kn this theorem was proved in a different way by Bollobas and Leader [77]; the general case was settled by Ahlswede and Bezrukov [3]. The
preceding proof is extracted from a more general approach, called Variational Principle by Bezrukov [55]. The graphs considered in the last two theorems are not regular (if not k1 = = kn = 1), hence we cannot derive a solution of the edge-isoperimetric problem. Moreover, this problem is here much more difficult because one has not always a nested structure of solutions; see Bollobas and Leader [77]. It is an interesting phenomenon, however, that omitting the bounds for the components yields an NSS; see [3]. In Theorem 8.3.1 we have already found ideals of given size with maximum weight. Now we study an analogous problem. Which antichain of given size generates an ideal of minimum weight? First we present a structural result that may be applied to S (which is selfdual by the succeeding Proposition 8.4.1) and to the dual pair T and Col. For S, the theorem was obtained by Clements [107], who significantly generalized preliminary results of Kleitman [302] (see Theorem 4.5.3(b)) and Daykin [123].
8.3 Optimization problems for Macaulay posets
363
Theorem 8.3.5. Let P and P* be graded, little-submodular, and shadow-increasing Macaulay posets. Let w : {0, ... , r(P)} -+ I[8+ be increasing, and let x :
P -> R+ be defined by x(p) := w(r(p)); that is, x is constant on the levels of P. Let 0 < m < d(P), and let 21(m) be the class of all Sperner families in P of size m. Further, let %I (m) be the subclass of Sperner families from 2t(m) that generate ideals of minimum weight. Then there are integers i and 1 < a < W1 such
that C(a, N1) U (Ni-1 - A(C(a, Ni))) E 211(m)
Proof. For any Sperner family F, let I (F) be the ideal generated by F. Let F E 2t1 (m), and let F' := CF. Recall that by (8.29) for all i
A,i (F') c CA,i (F) This implies
x(I(F')) _
w(i) IA,i (F')I < i
w(i)I A_i (F) I = x(I (F)) i
Consequently, F' also belongs to %I (m). Let 212(m) be the class of all canonically compressed Sperner families from %I (m). We saw earlier that 212 (m) is not empty. Let 213 (m) be the class of families from 212(m) for which r(I(F)) = EpEI(F) r(p) (r is the rank function) is minimum. For any F, let 1(F) := min{i : f,, # 0} and u(F) := max{i : f 01. We write briefly 1 and u if F is clear from the context.
Claim 1. For anyFE213(m),A_,t(F)=Ni,orFcN1andA(F)=Nt_1. Proof of Claim 1. Assume the contrary and let i be the largest index for which
A,i (F) = Ni (possibly i = -1). By our assumption, i < 1. Since A--,i+ I (F) is compressed and not equal to Ni+1, the last element p of N1+1 (with respect to - r (p),
since otherwise F C Nt, A(F) = N1_1. Now, F': = (F - {q}) U (p) is a Sperner family. Let F" := CF'. Then
x(I(F"))
x(I(F')) = x(I(F)) -x(q)+x(p) = x(I (F)) - w(r(q)) + w(r(p)) < x(I (F))
Consequently, F" E 212(m). But in the same way we derive r(F") < r(F), and this is a contradiction to F E 2t3(m).
Claim 2. For any F E 213(m), u(F) -1(F) < 1. Proof of Claim 2. Assume the contrary, and let F E 213 (m) with u -1 > 2.
Macaulay posets
364
Case 1. I A (Fu) I < I F1 1. We will show that there exists a canonically compressed
Sperner family F' with parameters
f':=
0
ifi > uori 2). In particular we have (for u - l > 2) A 1+2(F) = A,1+2(F'). Let X := N1+1 - A,1+1(F). In order to see that the construction works until level 1 + 1, we must verify that IXI > f,. Assume the contrary, I X I < fu. By the construction of a canonically compressed Sperner family we have F1 c Anew (X) (moreover, we have equality because of Claim 1). Let X' be a compressed subset of Fu of size IXI. From Proposition 8.1.4 and Proposition 8.1.6 we conclude IA(Fn)I >- IA(X')I >- IA(CX)I = IAnew(CX)I >- IAnew(X)I >- IF1I This is a contradiction since in our case I A (FF,) I < I F1 I. Finally, we must verify that also F1 can be constructed. We have by construction Fl+1 = Fl+t UY (UA (Fu)
if u - 1 = 2), where Y consists of the next fu elements after the last element of F1+1 (with respect to - 1 + 1 and fu > 0. This is a contradiction to F E 2t3(m). Case 2. I 0 (Fu) I > I F1 I. Let X be the family of the last f elements of 0 (Fu) (with respect to -1+1(F) (F - Fl - Y) U X U Z is a desired family. We with I Z I =IYI since then F' have
IN1+t - 0-±i+t(F)I 2 IV(Ft)I.
(8.32)
Note that F1 is a final segment in N1 (with respect to -- IV(CX)I > lonew(X)I.
(8.33)
Vnew(X) 2 v(X) n Fu
(8.34)
We have
since V(Nu-t - A(Fu)) n Fu = 0, and Vnew(X) = V(X) - V(Nu-t
A(Fu)) (note that Ni_1 - A(F,,) is a final segment). The relations (8.32) to (8.34) imply
IN1+1 - 0-,1+t(F)I >- IV(X) n F'ul = IYI, and thus the family Z can be found. Let F" := CF'. We have
x(I(F")) 5 x(I(F'))-<x(I(F))-IYIw(u)+IYIw(1+1)5x(I(F)), and
r(I(F")) = r(I(F')) < r(I(F)) since u > 1 + 1 and I Y I > 0. This is a contradiction to F E 213(m). With Claim 1 and 2 the theorem is proved.
Macaulay posets
366
We note that, for Theorem 8.3.5, we do not need little-submodularity completely. It is sufficient to suppose the first inequality in (8.19) for P and P*. Corollary8.3.1. Let P and P* be graded, little-submodular and shadow-increasing Macaulay posets. Then P and P* have the Sperner property. Proof. Let m := d (P). By Theorem 8.3.5, there exists a Sperner family F of size m such that for some i, F CN;_1 U Ni, A := F fl Ni is an initial segment and B := F fl Ni _ 1 is a final segment. It is sufficient to show that m < max { W1_ 1, W;).
Assume the contrary. Then l A (A) I < IAI since otherwise IFI = JAI + IBI < Ii(A)I + IBI < Wi-1. Analogously, IV(B)I < IBI. Case 1. IAI < I Ni - A 1. Let A' be the set of the first IAI elements of N; - A and let F' := F U A' - tnew(A'). Then F' is obviously a Sperner family and by Proposition 8.1.4,
IF'I = IFI + IA'I - Ionew(A')l >- IFI + IAI - lonew(A)I > IFI, a contradiction. Case 2. 1 A I > I Ni - A 1. Let A' be the set of the last I N1- A I elements of A and
let F' :_ (F - A') U Anew(A'). Again, F' is a Sperner family and by the second inequality in (8.19),
IF'I = IFI + I Anew(A')I - IA'I >- IFI + Ionew(Ni - A)I - INi - Al
= IFI + IBI - IV(B)I > IFI, a contradiction.
Theorem 8.3.6. The numbers i and a in Theorem 8.3.5 are uniquely determined
if all weights are positive. We have i = mini j : m < Wj) and a = min{b : b + Wi-1 - I A(C(b, N1))I = m}. Proof. First note that i is well defined by Corollary 8.3.1. Let Wh be the largest Whitney number. Then i < h. In order to see that a is also well defined, let
g(b) := b + Wi-1 - I A(C(b, Nt))I
We have g(O) = W1 -I < m < g(Wi) = Wi. Moreover, g(b + 1) - g(b) < 1 since IA(C(b, N1)) I cannot strictly decrease. Thus there is some b with g(b) = m and hence a is well defined. The Sperner family F := C(a, N1) U (Ni_1-A(C(a, Ni))) has size m, and we have i-1
w(I (F)) = E w(j)Wj + w(i)a. j=o
(8.35)
8.4 Further results for chain products
367
Let F' be a Sperner family of size m that generates an ideal of minimum weight and that has the form from Theorem 8.3.5:
F' = C(a', Ni') U (Ni'-t - O(C(a', Ni,))). Claim 1. We have i' > i. Proof of Claim 1. Assume the contrary. We consider the [0, i']-rank-selected subposet Pio,,,1. Clearly, F' is a Sperner family in P[o,,!]. Moreover, P[o,i,] satisfies
all conditions of Corollary 8.3.1; hence it has the Sperner property. Since i < h and because of Proposition 8.1.6, Wo < ... < W,'. Consequently, m = I F'I < W,' < m, a contradiction.
Claim 2. We have i' < i. Proof of Claim 2. Assume the contrary. Then
i'-I
w(I(F')) =
i
w(j)Wj + w(i')a' > E w(j)Wj > w(I(F)), j=0
j=0
a contradiction.
Now we know that i' = i, and it remains to show that a' = a. Obviously,
w(I(F')) - w(I(F)) = w(i)(a' - a). Consequently, a' < a. But a' < a and the definition of a imply IF'I # m, a contradiction. We conclude this section with some remarks: Sturmfels, Weismantel, and Zieg-
ler [448] used the preceding results (for S) to bound the cardinality of reduced Grbbner bases of n-dimensional lattices in Z". For the Boolean lattice, Labahn [331] determined the maximum size of Sperner families having a lower shadow of exactly m elements in the ith level. The solution of the vertex-isoperimetric problem for B" (see Theorem 2.3.3(b)) can be generalized in a natural way to the Hasse graph of chain products. This was proved by Bollobas and Leader [76]. Chvatalova [102] (n = 2) and Moghadam [372] solved earlier the related bandwidth problem for this graph (cf. Theorem 2.3.5). Again, see the forthcoming book of Harper and Chavez [261 ].
8.4. Some further numerical and existence results for chain products Until the end of this chapter, let S := S(kt, ..., k"), s := r(S) = kl +
+ k". We may assume, w.l.o.g., that k" 1. Let -< always be the reverse lexicographic order on S. We recall that the rank-generating function of S is given by
Macaulay posets
368
F(S; x) =
x+
+ xk'). For example, in S(4, 3, 2) we have the
following Whitney numbers: 9
E Wixi = (1 + x + x2 + x3 + x4)(1 + x + x2 + x3)(1 + x + x2) i=o
= 1+3x+6x2+9x3+11x4+11x5+9x6+6x7+3x8+x9.
We know from Theorem 8.1.1 that
min{IA(F)I : F C Ni, I FI = m} = IO(C(m, Ni))I So it is worthwhile to look for an algorithm that computes I A(C(m, Ni))I. For brevity, we use the notation
(J) s:=
Wi(ki,...,kj)
if 1 < j 0 assume that F contains all elements of Ni ending with 0, an, 1, an, ... , an_1 - 1, an but not all elements ending with an _ 1, an (an _ 1 < kn _ 1); that is, m' < (i Qi) s. The number of the first n-2 1 mentioned elements equals (n-2) n-2 1-an s, -an_1 s, ... > (.1-an-an-1+1 s' In the next
step the algorithm determines jan+1 := n - 2 since (i n)s -< m' < (i an)s' jan+t_1 = n - 2, 2 < t < an-1. Suppose we know already that jan+1 Then in the (al + t)th step the algorithm determines jan+t := n - 2 since ((n -2 M
s+...+
i -an
but
n-2
n-2
i -an -t+2 s
i -an -t+1
n-2
s
(_zi+1)
i-a-t+2
1-an S
n-2
>
s
Again the last inequality is true because otherwise in view of (8.36)
n-2 1-an + >
s
n-2 -an-t+1
n-2 l'-an-t+2 s+...+
s
n-2 an-t+lkn-1
s
n-1 f
S
Thus Jan+') an S +
+ ( _Jan+an-' an-an-1+1) S counts exactly the members of F having last coordinates 0, an, 1, an , ..., an -I - 1, an. Now we may continue in the same way, looking for members of F (which is compressed) having last coordinates an_ 1, an and classifying with respect to the third last coordinate and so on. If F # N, , (a 1, a2, ... , an) is the last element of F and if t is the largest index such that F contains all elements of Ni ending with at, ... , a, (note that t > 2), then induction (the first steps discussed earlier) easily yields the sequence of the numbers j determined by the algorithm: 1
1
n - 1,...,n - 1,n - 2,...,n - 2,...,t - 1,...,t - 1. an
an-1
a1+1
Thus (a) and (b) are proved. For (c), it is sufficient to observe that A (F) contains exactly all elements of Ni_ 1 with last coordinate 0, 1, . . . , an - 1, with last two coordinates 0, an, 1, an, ... ,
8.4 Further results for chain products
371
an- 1, an, and so on; thus
n-1 (i - 1)s
+...+
n - ln -2
\i -an)s+ i -an - 1)s n-2an-
C
1 - an -
t-1
) i
1 - an - ... - at - 1
3
) s
elements.
We emphasize that with the algorithm one may compute also the last vector of an m-element compressed family in Ni with respect to -: The numbers , a1+1, at + 1 can be obtained by counting the numbers j equal to n - 1, an, n - 2, ... , t - 1. Clearly (a1, ... , at-1) is the last vector in N;_an_..._at (k1, ... , kt_1), and thus we take at-1 as large as possible, then at-2 as large as possible and so on up to a I. The representation (8.37) of the number m with the properties of (b) in Theo-
rem 8.4.1 is called (as the algorithm) the i-representation of m; see also p. 47. It is not difficult to see that this i-representation is unique (cf. [109]). Example 8.4.1. Computation of the size of the shadow.
Consider N7(4, 3, 2) and m := 5. Note that F = {430, 421, 331, 412, 322} is compressed and 0(F) = {420, 330, 411, 321, 231, 402, 312, 222}. The Whitney numbers (l)s are given in the following table.
j=3 j=2 0
1
1
3
2
6
3
3
9
4
4
11
4
5
11
3
6
9
7
6
j
2 1
n n 0
2
0
n
0
The algorithm determines ji := 2, j2 := 2, j3 := 1, j4 := 1, jg := 1, that is, 5
-
(72)s
(62)s
+
(4I)s
(51)s
+
+
(3I)S.
+
For the size of the shadow, we obtain 8
-
(52),
(62)s
+
(4I) s
+
(1)
(3I) s
+
+
2 s
Macaulay posets
372
An analogous study of the i -representation of a number m for the poset T (k, ... , k)
was done by Clements [113] and by Leck [333]. The technical details are more involved. We have now an explicit formula for the minimum size of the shadow, but this formula is sometimes difficult to handle. Thus we will present easier estimates of the shadow. First recall that by the normality of S and in view of Proposition 4.5.2
IV(F)I >
IFI
Wi+1(S) - Wi(S) IA(F)l > IFI
f or a ll F C Ni,
i = 0 , ... , s - 1
for allFCN1,
i = 1 ,..., s .
-
,
(8 . 38)
(8 . 39)
Wi- I(S) - Wi(S)
From the rank symmetry and rank unimodality of S (cf. Proposition 5.1.1), we infer immediately
s-
IV(F)I > IFI for all F c Ni,
i
IFI for all F C Ni,
i>s
(8.40)
2
1.
(8.41)
2
But Theorem 8.1.1 gives for "small" sizes of F a "best" normalized matching inequality:
Corollary 8.4.1. Let F C Ni(S(kl, . . . , kn)) and let a = (al, ... , an) E S be > an and IFI < W1(S(al,... , an)). Then the first vector in S such that al >
IA(F)l
-
Wi-1(S(al,...,an)) IFI.
Wi(S(a1, . . . , an))
Proof. W e must show that CF belongs to S(ai,... , an) since then A(CF) belongs to S(al, ... , an), too, and the assertion follows from Theorem 8.1.1 and (8.39). So assume the contrary. Let i be the largest index such that there is some
b = (bl,...,b,) E CF withbi > a. Then CF ¢ S(ki,...,ki_l,ai,...,an); that is, IFI = ICFI > Wi(S(kl,... , ki-1, ai, ... , an)) > W1(S(at,... , an)), a contradiction.
Using a shifting technique similar to the proof of Lovasz's theorem (Theorem 2.3.1) BjSrner, Frankl, and Stanley [66] found the following estimation for S(oo, ..., oo) which we present without proof: Theorem 8.4.2. Let F C Ni (S(co, ... , oo)), IFI = (x), x E R, x > i > 1. Then
IA(F)l >- ji) By the way, Leck [333] used independently the shifting technique for another proof of the Clements-Lindstrom Theorem.
8.4 Further results for chain products
373
In Theorem 8.3.6 we provided formulas for the determination of i and a which are necessary for the "ideal minimization." For S, we will present an algorithm that computes a. In the case of B,,, Daykin [123] found an explicit formula for the minimum size of an ideal generated by an m-element Sperner family (and, more generally, m-element k-family). Clements [108, 111] also settled the case
of S. First we need a representation of a number m' which is similar to the i representation. Let L i JS
\i/s - \i J 1/s
Algorithm. i-difference representation of m'.
Input: kl,...,kn,i,m' > 0. Put 1 := 0; Repeat
Put 1:=1+1; Let Ji be the largest integer in [n] such that the following conditions hold:
[ii], (ii) j,
(i) m' >
j1-1 (if 1 > 1), (iii) ji < ji-+i (if 1 >
S
Putm'm'- ails; I
Put i:=i-1
L
until m' = 0.
Output: jt,j2, ,j1 If the algorithm terminates, we have a representation of m' in the form
m'=[i]s+[i-2l]s+...+[i-11+1]s
(8.42)
Theorem 8.4.3. Let 0 < w(0) < ... < w(s), and let x : S -* R be defined by x(a) := w(r(a)). Let (i"I)S < m < (i)s i < 2. Then the algorithm i difference representation of m' terminates for m' := m - (i n 1) s, m' > 0, and
yields a representation (8.42). The minimum weight of an ideal generated by an m-element Sperner family equals
i-I
E
w(j)(n)s+w(i)((i1)s+...-}-(i1+1)s/
Proof. The claim is essentially a reformulation of Theorems 8.3.5 and 8.3.6 (specialized to S). We have to show only that the algorithm terminates and that a = (,')s + + (i_i+I)s. The case m = (".)s is trivial; thus let m < (")S. Then,
Macaulay posets
374
clearly, jl < n - 1. Let e be the last element in C(a, N1). Moreover, let t be the largest index such that (el, ... , e,_ 1) is the last element in Nj_en _..._e, (S(k1, ... kt_1)). From the proof of Theorem 8.4.1 we know that n - 1 ,
.
.
.
, n - 1 , n - 2 ,
en
.
.
.
, n - 2,...,t - 1,...,t - 1
,
(8.43)
e,+1
en-1
is the j-sequence for the i-representation of a. For the proof, it is enough to verify that the algorithm i-difference representation of m' determines exactly the same j-sequence. For brevity, we use the notation h(b)
IC(b, N;)I - I A(C(b, N1))I = b - I A(C(b, N1))I.
Note that h (a) = m' and a is the smallest natural number with this property. Let us assume that the algorithm i-difference representation of m' determined already the numbers n - 1,...,n - 1,...,u - 1,...,u en
V
in the right way, that is, as in (8.43) and with u > t. We have to show that the algorithm terminates in the next step with the number u - 1 if u = t and v = e and that otherwise the next number is
u-1 ifv<e,,, u-2 ifv=e,,,u>t, andeu_1 > 0, u-z
ifv = eu, u > t, and eu_I =
= eu_z+2 = 0, eu-z+1 > 0.
Case 1. v < e or u = t and v = eu. Let (fl, ... , f,_ 1) be the last element in and let f := (fl, ..., fu-1, v, eu+1, ..., en). N'-en-..._e,,+i_v(S(kl, ..., Let f be the bth element in Ni with respect to {. Then b < a and by definition of a, h (b) < h (a). Consequently,
h(b) =
Ln-11 I
i
S
+...+L
i-
u-1
+...+I
+Ln-11
Js
S
u-1
1
<mr
Thus, if v < e,, (
0 (otherwise we have to replace u - 1 by u - z and to add in the - e and following vector f correspondingly more zeros). Let it := i - e fu-2) is the last element in A-2, 0, e....... et) where (fl , let f = (fl , -
-
N1'(S(ki, ... , k, -2)). The arguments from Case 1 show also in this case that the next element in the j-sequence in the algorithm i-difference representation of m'
is at least u - 2. We only must prove that the next element cannot be u - 1. This is clear if e = k (see condition (iii) in the algorithm). Thus let e < ku.
Let g = (gi, ..., N1 , (S(kl , ... ,
e,,, ..., et) where (gi, ..., gt) is the last element in 1)). Let g be the bth element in Ni with respect to - 0, and equality means that the algorithm has already terminated. The second case yields the desired inequality (8.44).
We conclude this section with a further existence theorem. For an element a = (at,...,an) E S, let a` := newelement iscalled the complement of a. For F C S, let F` := {a` : a E F) (the complementary family).
Proposition 8.4.1.
(a) The mapping a H ac is an isomorphism between S(ki, ..., k,) and its dual; thus chain products are self-dual.
(b) If a < b then b` < a`. We omit the trivial proof. The following theorem of Daykin, Godfrey, and Hilton [125] (for (resp. of Clements [105] (for S)) says that with each Sperner family in S we may associate a "reflexed" Sperner family. The result had been conjectured by Kleitman and Milner [309].
Theorem 8.4.4. If f is the profile of a Sperner family in S then so is f', where
f;'=
0
ifi2. Proof. We assume s to be even. For s odd, the arguments are analogous. Let m be the largest integer for which ff+m + ff_m > 0. We proceed by induction on m. If m = 0, we may take F' := F. Now consider the step m - 1 --* m where 1 < m < s/2. Let F' := CF (recall that Ft is the canonically compressed Sperner family with profile f which exists by Theorem 8.2.3). By the construction of F, we have
(F'),+m = C(.ff+m, Let
F2 :_ (F' - C(.f +m, N, +m)) U A(C(f +m, N +m)) Since F' is a Sperner family, F2 is a Sperner family, too. Next let us turn to the complements. Let F3 := (F2)`. We again replace F3 by the canonically
8.4 Further results for chain products
377
CF3. As for F1 we have
compressed Sperner family F4 (F4)7,
+m = C(f -m, N,+m)
Let
F5 :_ (F4 - C(f -m, Ni+m)) U A(C(f-m, N+m)) For the parameters fo , f5
,
f5 of the Sperner family F5, we have
fs-1 + I A(C(f -m,
ifiZ+mifi=2-m+1, ifi=2+m-1,
fs-i
if2-m+1 - I(F3)7I = ff
(8.47)
8.5 Spemer families satisfying additional conditions
379
Let F4 := (F3 - (F3),s12) U 0((F3)s/2). Then F4 is again a Sperner family, and its parameters are given by
f4 =
ff + fs-i
if i < 1 - 2,
fl, -1 + ff+t + IA((F3), )I
if i = i - 1, ifi > 2.
0
By (8.46), (8.47), and in view of the rank symmetry,
Sf E -' f,. _
1 > i=o
Wi -
;o
i
Corollary 8.5.1.
I
Wi
IA((F3))I
+
S
WS-1
f
i=o Wi
i#I
fs R'L
J
The maximum size of a complement free Sperner family in S
equals W,s-,i Proof. The upper bound follows from Theorem 8.5.1 and the rank symmetry and
rank unimodality of S. Clearly, N,l is complement free; that is, the bound is really attained. Note that Clements and Gronau [116] and Gronau [161] determined in several cases all maximum complement-free Sperner families.
A family F C_ S is called self-complementary if F = F'. In the Boolean case ki = k = 1 we know much about self-complementary Sperner families because of the Profile-Polytope Theorem (Theorem 3.3.1) and Remark 4 after it. In order to present some results for S we need some further notations. Let t := 0 if ki, . . . , k, are even, and otherwise let t = t (S) be the largest index such that kt of S as is odd. W e define, f o r max{t, 1 } < i < n, subsets S = 5 ' ( k 1 . . . . . follows:
S; := I a E S : ai
m (C(F+)) l = IO-*m (C(G`)) I < I tX-,m (G`) 1.
If c is the last vector of O,m (G`) then clearly a -< c. Moreover, c -< b` since b` is the last vector of G`'. Hence a -< b`.
The relations a -< b, a < b` imply a j < k j /2 for all j = 1, ... , n and a; < ki/2 for some 1 < i < n. Obviously, the largest i with this property satisfies
i > t. Thus n
C(F+) c U S;
.
max(t,1 )
Obviously,
/
S; =SI k1,...,ki-l, [k12 1]),
(8.49)
8.5 Sperner families satisfying additional conditions
381
kj. Moreover,
but note that the minimal elements in SJ have rank hi := 2
the sets Si are pairwise disjoint. If F is a self-complementary Sperner family, then by Theorem 8.5.2, the sets Gi := C(F+) fl S* are pairwise disjoint antichains whose union is C(F+). Let & 0, ... , gi,s be the parameters of Gi. Note that
gi, j = 0 if j < 2 and that F-"=max(t,
or
j > s - hi-1,
(8.50)
I) gi,j, j = 0, ... , s, are the parameters of F+. The LYM-
inequality (8.46) in each S! yields
1 and F- = 0. Since I F I = I F+ I + I F- I and I F+ I = IF-1, it is enough to show that each G1 has size at most W(S+1)/2 (S; ). Recall the isomorphism (8.49). The Whitney numbers Wj (S* ) are decreasing for 2 < j < s - h i _ 1 since S is rank symmetric and rank unimodal,
and the largest level of S! is at rank h i + L (kl + Because of (8.50) and (8.51) it follows that 2 IGdd =
57
gr,j
W.
+ ki _ 1 + 2 J ) < 2 (S,*).
P < j<s-hi+1
That this bound is the best possible can be easily seen by taking F := N., if s is even and F := G U GC where G := U"_tNs (Sr) if s is odd.
We say that a family F in S is dynamically intersecting (resp. dynamically cointersecting) if for all x, y E F there exists an index i such that x; + y; > k; (resp. xi + y; < k;). In the Boolean case kl = k = 1 these notions coincide with the notions intersecting (resp. cointersecting). The following propositions are obvious.
Proposition 8.5.1. A family F is dynamically intersecting iff its complement Fc is dynamically cointersecting. Proposition 8.5.2. If the family F is dynamically intersecting then it is complement free.
These propositions and Corollary 8.5.1 immediately imply:
Theorem 8.5.4. The maximum size of a dynamically intersecting (resp. dynamically cointersecting) Sperner family in S equals WLJ.
Proposition 8.5.3. Let F be a dynamically intersecting and cointersecting Sperner family. Then F fl F` = 0, and F U Fc U So is a self-complementary Sperner family, where So is given by (8.48).
Proof. The disjointness of F and F` follows from Proposition 8.5.2. Clearly, F and F` are Sperner families and F U Fc is self-complementary. Moreover, no element x of F (resp. F`) is related to (z , ... , 2) because otherwise the pair x, y with y := x cannot satisfy the dynamic intersecting as well as the dynamic cointersecting condition. Hence it remains to show that there are no x E F, Y E F` such that
xy.
8.5 Sperner families satisfying additional conditions
383
Assume the contrary. We have y = z° for some z E F. Thus, for all i, xi < ki - zi, or, for all i, xi > ki - zi. In the first case F would not be dynamically intersecting, in the second case it would not be dynamically cointersecting, a contradiction.
Theorem 8.5.5. family, then
If F is a dynamically intersecting and cointersecting Sperner
IFI
0,
2 (W7g - 1)
if s is even and t = 0,
=t Wsi (S!)
ifs is odd,
and the bound is the best possible.
Proof. The bound follows from Proposition 8.5.3 and Theorem 8.5.3. To see that it is the best possible take, for even s, from each pair of different complementary elements of Ns12 exactly one member, and, for odd s, the set G from the end of the proof of Theorem 8.5.3. In the light of the Erdo"s-Ko-Rado Theorem we study also the restriction to one level (here in the cointersecting case).
Theorem 8.5.6. Let F be an 1-uniform dynamically cointersecting family in S. Then IFI 2. In this case F is automatically also dynamically intersecting, hence, by Proposition 8.5.3, F U F` is a self-complementary Sperner family (not containing (2 , ... , )), and, by Theo2 rem 8.5.2, n
CF C
U S. i=max(t,1 }
Consequently, n
IFI = CFI < E Wt(SI ) i=max{t, 1)
The family bound is the best possible.
is obviously dynamically cointersecting; that is, the
384
Macaulay posets
In Section 7.3 we already studied statically t-intersecting families (see also Theorem 3.3.4). Here we investigate the case t = 1 repeating and extending the definition. A family F in S is called statically intersecting (resp. statically cointersecting) if for all x, y E F there exists some coordinate i such that x; , yi > 1 (resp. xi, yi < ki - 1). Recall also the definition of the support as supp(x) :_ {i E
[n] : xi > 11, supp(F) := (supp(x) : x E F). The cosupport is defined similarly:
cosupp(x) := {i E [n] : xi = ki}, cosupp(F) := {cosupp(x) : x E F}. As in Proposition 3.3.1 we have:
Proposition 8.5.4. The family F C_ S is statically intersecting (resp. statically cointersecting) iff supp(F) (resp. cosupp(F)) is intersecting (resp. cointersecting).
The following propositions are obvious and need no proof.
Proposition 8.5.5. A family F C S is statically intersecting iff its complement F` is statically cointersecting. Proposition 8.5.6.
(a) If x, y E S satisfy the statically intersecting (resp. cointersecting) condition, and if z > x (resp. z < x), then z, y also satisfy the corresponding condition.
(b) If r(x) + r(y) > s (resp. r(x) + r(y) < s), then x, y satisfy the statically intersecting (resp. cointersecting) condition.
If ki = kn = 1; that is, if S is the Boolean lattice B, the statically intersecting (resp. cointersecting) condition coincides with the usual intersecting (resp. cointersecting) condition. In this case, the Profile-Polytope Theorem 3.3.1 gives us enough information about our families. In general, the statically intersecting and cointersecting conditions are more difficult to handle than the dynamic intersecting and cointersecting conditions. The reader may get an idea of the difficulties from the related Theorem 7.3.1. Only some partial results are presented here. In
particular, we restrict ourselves to the case k := kl =
= k,,. For even n,
the set
S :_ {x E N, : xi E {0, k}, i = 1, ... , n} plays an exceptional role. Obviously, BSI = G12)' The last theorem in this section we obtained together with Gronau in [161].
Theorem 8.5.7. Let k, = kn = k > 2.
8.5 Sperner families satisfying additional conditions
385
(a) If F C S is a statically intersecting (resp. statically cointersecting) Sperner family then
IFI 2n - 1.
(b) The same assertion is true if F C S is a statically intersecting and cointersecting Sperner family.
Proof. First we show that the bound can be attained. For odd n, let F := N15i21. Then, for all x E F, I supp(x) I > 2, which implies that F is statically intersecting. Moreover, we have for all x E F, I cosupp(x)I < 2 since otherwise (recall that n is odd), r(x) > n21k > 121. Hence F is also statically cointersecting. If n is even, S consists of pairs of complementary elements. We partition S into two sets Si and 32 by putting for each such pair one member into Si and the other into S2. Then ISiI = 1321 = 2(nj2) and supp(S1) (resp. cosupp(S1)) is intersecting (resp. cointersecting). Let F := N512 - 32. Then, for all x E F, I supp(x) I > 2, but I supp(x)I = 2 only if x E St. Hence I supp(F) I is intersecting; that is, F is statically intersecting. In the same way it follows that F is statically cointersecting. Hence (a) and (b) are proved if we can show that the upper bound for (a) is correct. Because of Proposition 8.5.5 we may restrict ourselves to the cointersecting case. For odd n, the upper bound is trivial since Wl5121 is the maximum size of a Sperner family by the Sperner property of S. Thus let n be even. The case n = 2 can be checked easily; thus let n > 4. For each family F, let 1 = 1(F) (resp. u = u(F)) be the smallest (resp. largest) index i such that f > 0. Let F be a maximum statically cointersecting family for which u -1 is minimum. It is enough to prove that I = u = 2 since for F C N512, there holds IF n SI < z ISI (from each pair of complementary elements of S at most one member may belong to F). Claim 1. We have u < '2' Proof of Claim 1. Assume u > 2 + 1. With the usual shifting we define the new family
F':=(F-Fu)UA(Fa) which is, in view of Proposition 8.5.6(a), still a statically cointersecting Sperner
family whose size is at least as large as the size of F by (8.41). Since u(F') I (F') < u(F) - l(F), we have a contradiction to the choice of F. Using the same arguments and Proposition 8.5.6(b) we obtain immediately:
Claim 2. We have l > 2 - 1. So up to now it is clear that F consists only of fs/2 elements of rank and A/2-1 elements of rank 2 - 1. We add one more condition on F: Of all maximum
Macaulay posets
386
statically cointersecting families in Ns/2-1 U N,12 we choose F such that fs/2 is maximal (it remains to show that A/2-1 = 0). Claim 3. We have
k, k - 1))
and
k))
f,' -1
n-1
n-1
Proof of Claim 3. Assume that f < Ws (S(k, ... , k, k - 1)). We consider the
n-l
canonically compressed Sperner family CF with the same profile as F. Because of the assumption, (CF), and A((CF) s) are subsets of S(k, ... , k, k - 1). But
n-l 22 - 1 = s - 1 = k - 1 + (n - 1)k. Hence, by (8.41) with s replaced by s - 1 IA((CF),)I > I(CF)2- 1.
Consequently (noting that CF is a Sperner family; that is, A((CF)s12) n (CF)s12-1 = 0), W,s-1 >-
Ii ((CF),)I + I(CF)2-ll I(CF), I + I(CF),-11 = ICFI = IFI.
However, consider the family F' = Fs FZ
_1 := {x E Ns-1(S) : x1 = k}
(8.52)
U F; where and
Fs' := Nz (S(k^ k, k - 1)). n-1
It is easy to see that F' is a statically cointersecting Sperner family. Since Ns
(S) = Ns-1(S(k....,, k, k - 1)) U FF-1 n-1
and
W _1(S(k,...,k,k- 1)) = Ws(S(k,...,k,k- 1)) n-1
n-l
it follows that
IF'I =
(8.53)
Hence, by (8.52) and (8.53), IF'I > IFI, and for the parameters we have
f
Ws(S(k...,k,k-1))> fs n-1
by our assumption. This contradicts the choice of F since A12 has to be maximal. Thus the first inequality is proved. In particular we know that (CF),12 contains
all elements of Nsl2(S) whose last coordinate is less than or equal to k - 1. By
8.5 Spemer families satisfying additional conditions
387
the construction of the canonically compressed Sperner family, every member of (CF),12_1 ends with k. Hence
k)) = Ws+1(S(k,.kk)).
f12 -t = (CF)s-11 < n-1
n-1
Claim 4. If k > n and 0 < i < (n - 1)k, then Wi(S(k,...,k)) < W1(S(k - 1,...,k - 1,n - 1)). n-1
n-1
Proof of Claim 4. Let, for 0 < a < n - 1,
Sa := S(k,...,k,k- 1,...,k- 1, a). n-a-1
a
Obviously, it is enough to prove Wi (Sa) < Wi (Sa+1) for 0 < a < n - 2, and this can be accomplished by constructing an injection cp from N, (Sa) into N, (Sa+1).
Here is such an injection: We put cp(x) := x if x E Sa n Sa+l and cp(x) :=
(X1,...,xn-a-2,xn-a-1-(a+l)+Xn,xn-a,...,xn_1,a+1)ifx E Sa-Sa+1 Since we have for X E Sa - Sa+1 the relations Xn < a < n - 1, Xn_a_ l = k > n, it follows that 0 < Xn-a-1 - (a + 1) + Xn < k - 1. Thus, indeed cp(x) E Sa+1 The injectivity can be easily verified, and r(x) = r(rp(x)) is obvious.
We conclude the proof of the theorem by showing that fs/2_1 > 0 yields a contradiction. With our assumption fs/2-1 > 0 there must exist some set I c [n] of minimum size such that G := {x E FF-1:cosupp(x) = I}
0.
Let
H := {x E V(G) : cosupp(x) = I}. Claim 5. The family F' := (F - G) U H is a statically cointersecting Sperner family.
Proof of Claim 5. By construction, cosupp(F) = cosupp(F'). Since F is cointersecting, Proposition 8.5.4 implies that also F' is cointersecting. The only obstacle to F' being a Sperner family could be the existence of some x E F -1- G and some y E H with x < y. But by the choice of I, I cosupp(x)I
cosupp(y)I =III and
Hence, x ¢ y. Claim 6. We have I F' J > I FI
cosupp(x)
1.
388
Macaulay posets
Proof of Claim 6. Since H fl F = 0 (F is a Sperner family), the assertion is equivalent to IGI < I HI. Let i := III. With each x E G U H we associate an element x' of S(k - 1, ..., k - 1) by deleting all coordinates xj of x for which n-i xi = k. Note that
r(x') = r(x) - ik.
(8.54)
We obtain families G' and H' with the obvious properties
IG'I = IGI,
IH'I = IHI,
V(G') = H',
where the upper shadow is considered in S(k - 1, ... , k - 1). Thus the assertion
n-i
is equivalent to
(8.55)
IG'I < IV(G')I. By (8.40) and (8.54), the inequality (8.55) is satisfied if
2(2 -1-ik)+1 < (n-i)(k-1), which is equivalent to
k> n - i i
-1
and this is true under our supposition k > 2n - 1 (i.e., k > n - 2) for i > 1.
It remains to consider the case i = 0. We turn to the complements in S(k - 1, ... , k - 1) and estimate the lower shadow. We have n
(GY C NZ+1-n(S(k - 1, ... , k - 1)).
(8.56)
n
Clearly, I (G')` I = I G' I = IGI < .fl, -1, and using Claim 3 and Claim 4 we obtain
(note 0 < 2 + 1 < (n - 1)k because of n > 4, k > 2) I (G')`I < WZ+1(Sn-1)
(8.57)
WZ+l(Sn-l) < Wi+1-n(Sn-1)
(8.58)
Moreover,
since(2+1)+(2+1-n)>n-1+(n-1)(k-1)byk>2n-1>n-2. From (8.56) to (8.58), we derive for the compression that
C((G')`) C Nj,+1-n(Sn-1), and using (8.41) and Theorem 8.1.1, we obtain
IV(G')I = IA((G')`)I >- IL(C((G')`))I > IC((G')c)I = I(G')`I = IG'I
8.5 Sperner families satisfying additional conditions
389
since 2(2:! + 1 - n) - 1 > n - 1 + (n - 1)(k - 1) is equivalent to k > 2n - 1. Hence (8.55) is verified.
Claim 5 and Claim 6 complete the proof of the theorem since contradicting the choice of F.
2
2
We conjecture that Theorem 8.5.7 remains true for even n and 2 < k < 2n - 1.
NOTATION
We omit in the following list those symbols that are very well known, selfexplanatory, or not used often enough to be worth listing.
Sets and Families N
nonnegative real numbers natural numbers (nonnegative integers)
[n]
set{1,...,n}
R+
k
2[n]
ACB ACB
A-B A
di,j Si.i (F)
E (F) X
